In 2004, Bogosian entered high school at Cushing Academy, a prep school in Massachusetts. The assistant coach of the school's hockey team was Ray Bourque, a former NHL defenseman who was elected to the Hockey Hall of Fame. Bogosian's teammates at Cushing included his older brother, Aaron, and Bourque's son, Ryan. While at Cushing he was used mostly as a fifth or sixth defenseman and played few minutes as a result. After his second year at the school, Bogosian was selected 19th overall in the 2006 Ontario Hockey League (OHL) Priority Draft by the Peterborough Petes Bogosian's decision to play in the OHL rather than go to the NCAA differed from most of his teammates, including his brother. As his father and uncle had also played college football in the NCAA, it was expected that Bogosian would follow them in going to university. He cited watching Ottawa 67's games at age nine as a major influence in deciding to play in Canada.
Bogosian began playing for the Petes in the 2006–07 season. He played 67 games for the team and finished with 7 goals and 26 assists for 33 points, which was second on the team in points for both rookies and defensemen. In recognition of his season, he was named to the OHL's Second All-Rookie Team, as one of the top four rookie defensemen in the league. The next season, Bogosian appeared in 60 games for the Petes. He led his team in scoring with 61 points (11 goals, 50 assists), the only defenseman in the league do so; the 50 assists he scored were second in the league among defensemen. The Petes reached the playoffs after missing the previous year, and Bogosian added another three assists in five games. During the season, he participated in both the OHL All-Star Classic and the CHL Top Prospects Game, an all-star game of prospects from the three leagues in the Canadian Hockey League (Western Hockey League, Quebec Major Junior Hockey League and OHL). At the conclusion of the season, Bogosian was named one of the five draft-eligible finalists for the Red Tilson Trophy as most outstanding player in the OHL, and was also named to the OHL First All-Star Team as one of the two best defensemen in the league.
Prior to the 2008 NHL Entry Draft, Bogosian was ranked by the NHL Central Scouting Bureau as the second best North American skater for the draft and the third ranked prospect out of the OHL. He was drafted third overall by the Atlanta Thrashers behind Steven Stamkos and Drew Doughty. He was noted by his coach in Peterborough, Vince Malette, to be a "very physical defenseman who is a complete player" and defensively solid who could skate well with the puck and not lose speed. Thrashers general manager Don Waddell liked the physical aspect Bogosian brought to a game, as well as his ability to help score goals while at the same time stop other teams' top players. Bogosian had been noted prior for the draft for his dedication to training, as he drove more than 90 minutes to Ottawa to get a better work out.
In July 2008, Bogosian attended the Thrashers prospect development camp. The youngest player of the 32 at the camp, he stood out as one of the best players there and effectively assured himself a roster spot on the team. On September 4, 2008, Bogosian signed a three-year, entry-level contract worth $2.625 million with the Thrashers. He was named to the team's opening day lineup, and made his NHL debut in the first game of the season on October 10 against the Washington Capitals. Bogosian became the youngest person to play for the Thrashers that night at 18 years and 87 days, surpassing Ilya Kovalchuk (18 years, 172 days). Twenty-four seconds into his first shift, Bogosian was called for a holding penalty, while later in the contest, he had a fight with Capitals enforcer Donald Brashear. Eight games into the season, however, on October 28, Bogosian broke his left leg in a game against the Philadelphia Flyers. Prior to rejoining the Thrashers, Bogosian was sent to their American Hockey League (AHL) affiliate, the Chicago Wolves, for conditioning. After playing five games with the Wolves, including scoring his first professional goal on January 3, 2009, against the Rockford IceHogs, Bogosian rejoined the Thrashers for the remainder of the season. He scored his first NHL goal and recorded his first assist in his twelfth game on January 17, 2009, against Pekka Rinne of the Nashville Predators. Bogosian finished his first professional season with 19 points in 47 NHL games and 1 goal in 5 AHL games as the Thrashers finished out of the playoffs.
Throughout the 2009–10 season Bogosian was bothered by an injury. During a game against the Ottawa Senators on October 31, 2009, Bogosian fell behind his team's net; with his left hand lying on the net's iron frame along the ice, another player fell on top of him. The impact broke his left thumb and injured his wrist. Though he did not miss any time from the injury, it had an effect on his offensive play; Bogosian was perceived to be more hesitant with the puck, choosing to pass it more often than shooting. In the first 17 games of the season, he had scored 8 goals, then went 29 contests without scoring. In the final 64 games, he only scored 2 goals. After he recovered, Bogosian admitted he did not play at his full capacity, but had decided not to disclose the injury to the team until the season finished in April 2010. Regardless of his injury, Bogosian only missed one game throughout the season, a game on March 14 against the Phoenix Coyotes. Even with these setbacks, Bogosian set career highs for games played (81), goals (10), assists (13) and points (23). The 10 goals he scored tied a team record for most by a defenseman in a season, first set by Yannick Tremblay in 1999–2000, the Thrashers' inaugural year. He also set a team record with 196 hits. Again the Thrashers missed the playoffs. After the Atlanta Thrashers were sold to True North Sports & Entertainment and were relocated to Winnipeg, Bogosian agreed to terms on a new two-year, $5 million contract with the Winnipeg Jets on September 14, 2011.
During the 2014–15 season, his seventh with the Thrashers/Jets franchise, Bogosian was traded alongside Evander Kane and the rights to Jason Kasdorf, to the Buffalo Sabres in exchange for Tyler Myers, Drew Stafford, two prospects and a conditional first-round pick in the 2015 NHL Entry Draft.
Bogosian made his international debut with the American national team at the 2009 IIHF World Championship in Switzerland. He appeared in all nine games for the team, and recorded one assist, in a game against France, as the Americans finished fourth overall. During the 2009–10 season he was mentioned as a candidate to join the American team at the 2010 Winter Olympics in Vancouver, but a wrist and thumb injury led to a decrease in performance and he was ultimately not selected. He was invited to join the American team at the 2010 IIHF World Championship, but declined in order to heal his injuries.
Bogosian was born to Ike and Vicky Bogosian in Massena, New York, a town of 13,000 adjacent to the Saint Lawrence River. He is the first NHL player of Armenian descent; his great-grandfather, Stephen Bogosian, left Western Armenia in 1923 at the age of 16 as a result of the Armenian Genocide. At the age of 2½, Bogosian first began to skate. He joined the Massena Minor Hockey Association when he was 3, staying in the program until he was 14, when he left for Cushing Academy. Bogosian first began to play defense at the Squirt level (under-10); though he was smaller than most of his peers, he was a good skater. He alternated between forward and defense throughout minor hockey. When he began attending Cushing Academy he was a forward the first year, before he converted to a defender full-time. After leaving the school and moving to Canada with two years of high school to complete, Bogosian took courses on the internet.
Zach has two older brothers, Ike Jr. and Aaron. Aaron is also a professional ice hockey player who attended St. Lawrence University where he played four seasons with the Saints. Aaron currently plays in the American Hockey League with the Springfield Falcons.
Ike was an athlete when he was younger, playing college football with Syracuse University from 1979 to 1981. In his senior season at Syracuse he was co-captain of the team with future New York Giants running back Joe Morris. He currently runs a carpet and floor cleaning business with his brother in Massena. Vicky works as a hairdresser.
He is married to Boston Breakers and former United States women's national under-23 soccer team midfielder Bianca D'Agostino.