Cogliano grew up in Woodbridge, Ontario, starting out playing hockey for the City of Vaughan Hockey Association and the Vaughan Rangers in the Ontario Minor Hockey Association (OMHA) before playing in the Greater Toronto Hockey League (GTHL) with the AAA Vaughan Kings.
Cogliano was drafted in the third round, 54th overall, by the Toronto St. Michael's Majors in the 2003 Ontario Hockey League (OHL) Priority Selection. Many believed that Cogliano was the top OHL prospect that year; however, due to his interest in playing collegiate hockey in the NCAA, he was drafted much later.
Cogliano spent two seasons with the St. Michael's Buzzers, a Tier 2 Junior "A" Team in Toronto. In his final season, he represented Canada at the Junior World Cup in the summer of 2004 and lead the team in scoring. He was also the League's leading scorer, as well as the only player from the Ontario Junior Hockey League (OJHL) to be selected in the 2005 NHL Entry Draft.
Cogliano then moved to the NCAA and played two years for the University of Michigan Wolverines. In 2005–06, he represented Canada in the 2006 World Junior Ice Hockey Championships in Vancouver, winning a gold medal. In 2006–07, he finished third on the Wolverines in scoring with 50 points in 38 games and again won gold with Canada at the 2007 World Junior Ice Hockey Championships in Sweden. He signed a professional contract on May 2, 2007, with the Edmonton Oilers, the team which drafted him 25th overall in 2005.
Leading up to the 2007–08 season, Cogliano had a strong training camp and made the Oilers' opening night roster. He played his first NHL game on October 4, 2007, against the San Jose Sharks, recording one assist. He scored his first NHL goal on October 8, 2007, against the Detroit Red Wings, beating All-Star goaltender Dominik Hašek.
On March 7, 9 and 11, 2008, Cogliano set an NHL record by scoring overtime goals in three consecutive games against the Columbus Blue Jackets, Chicago Blackhawks and St. Louis Blues, respectively. His stick and gloves were subsequently sent to the Hockey Hall of Fame.
Cogliano would play in all 82 of Edmonton's games during his rookie season, leading the team in appearances. He would score 45 points (18 goals and 27 assists), leading the Oilers with five game-winning goals while recording the team's highest shooting percentage at 18.4%.
In his second NHL season, his statistics declined slightly, along with fellow Oilers sophomore Sam Gagner. Cogliano posted 38 points (18 goals and 20 assists) while again playing in all 82 of his team's games. That season, Cogliano was invited to participate in the NHL YoungStars Game during the All-Star weekend in Montreal, playing for the Sophomores team. Cogliano also participated in and won the "Fastest Skater" competition during the SuperSkills event, clocking in at a time of 14.31 seconds.
In June 2009, Cogliano was confirmed to be a part of a planned blockbuster trade that would include Ladislav Šmíd and Dustin Penner being sent to the Ottawa Senators in exchange for the disgruntled All-Star forward Dany Heatley. In the end, however, Heatley exercised the no-trade clause contained in his contract and the deal could not be finalized. Heatley would shortly after accept a trade to the San Jose Sharks instead, and Cogliano remained an Oiler.
The subsequent 2009–10 season was a difficult one for Cogliano, as his offensive production again declined and his team struggled. He would finish with 28 points (10 goals and 18 assists) in 82 games, and the Oilers finished last overall in the NHL. He was tried at all three forward positions during the season.
By the end of the 2010–11 season, Cogliano was the only Oiler who played in all 82 games, and was fifth in team scoring, with 35 points.
As part of the Oilers' rebuilding process, Cogliano was traded to the Anaheim Ducks on July 12, 2011, for a second-round draft pick in the 2013 NHL Entry Draft.
In his second year with Anaheim, and his sixth season overall in the NHL, Andrew made his debut in the Stanley Cup playoffs, where the Ducks, as the second seed in the Western Conference, were eliminated in the first round by the seventh-seeded Detroit Red Wings in seven games.
During the 2013–14 season, Cogliano played his 500th consecutive game in a home game against the San Jose Sharks on New Year's Eve. He became the 20th player in NHL history to reach that milestone, and only the fifth to do so from the beginning of his career. On January 25, playing in Dodger Stadium against the Los Angeles Kings in a NHL Stadium Series game, Cogliano scored into an empty net for the 100th goal of his NHL career. He finished the season with 21 goals, setting a new career high. It was his first season with 20 or more goals. In the playoffs, he scored his first-ever playoff goal against the Dallas Stars.
Cogliano grew up in Woodbridge, Ontario. He has a brother, Matthew, who is an elementary school teacher.