Beginning in 2000–01, Kostitsyn spent the first two seasons of his early career with Polimir Novopolotsk, competing in several leagues, including the Eastern European Hockey League (EEHL). In the summer of 2002, he signed with CSKA Moscow and moved to Russia. He split his time with the organization between CSKA 2 of Russia's developmental league and also continued to play in the Belarusian league with Yunost.
After his first season with CSKA, he was selected by the Montreal Canadiens in the first round, tenth overall, of the 2003 NHL Entry Draft. Kostitsyn drew favourable scouting reports from his strong showing at the 2003 World U-18 Championships and was expected to be taken even higher in the draft; it is speculated that health concerns with his back caused teams to pass him over.
In the summer of 2004, he was invited to the Montreal Canadiens' training camp and signed a three-year deal with the club. He was subsequently assigned to the Canadiens' American Hockey League (AHL) affiliate, the Hamilton Bulldogs, for the 2004–05 season. He scored his first goal with Hamilton in his first game, on 15 October 2004, and finished the season with 23 points in 66 games.
Midway through the 2005–06 season, he was called up by the Canadiens and played his first National Hockey League (NHL) game on 1 December 2005, but was limited to just one shift in an overtime loss to the Buffalo Sabres. He was called up again later in December and scored his first career NHL goal on 13 December against Curtis Joseph in a 5–2 win over the Phoenix Coyotes.
Kostitsyn had a breakout season in 2007–08, playing on Montreal's most productive line alongside Alexei Kovalev and Tomáš Plekanec. He amassed 53 points in 78 games and was joined by younger brother Sergei, who had been called up from Hamilton, midway through the season. Entering the playoffs with the Canadiens as the first seed, Andrei and Sergei both scored in their NHL post-season debuts against the Boston Bruins less than two minutes apart. They also both finished with eight points each in 12 playoff games. However, the Canadiens were defeated by the Philadelphia Flyers in the second round. On 1 July 2008, Andrei signed a three-year extension worth an annual salary of $3.25 million with the Canadiens.
Several games into the 2008–09 season, he suffered a concussion when he was checked into the boards by defenceman Kurt Sauer of the Phoenix Coyotes. Kostitsyn lay on the ice for several minutes before needing to be helped off the ice. He returned to the team after missing two games. In December, he missed another two games due to a minor leg injury, then returned to notch his first career NHL hat trick on 27 December 2008, in a 3–2 win against the Pittsburgh Penguins.
On 27 February 2012, Kostitsyn was traded to the Nashville Predators for a 2nd round pick in 2013 and a 5th round pick in 2013.
On 1 May 2012, Kostitsyn and Predators' teammate Alexander Radulov were spotted at a Scottsdale bar at 5:00 AM the night before Game 2 of their second round playoff series against the Phoenix Coyotes, thus breaking curfew. Both were subsequently suspended by Predators' management for Game 3. Said Predators' general manager David Poile of the incident: "What they did was unacceptable and the coaches and myself had to come to the plate and do the right thing for the team. It happened. It's really unfortunate. It's selfish behaviour and we'll just have to leave it at that."
During the 2012 offseason, the Nashville Predators have made it evident that they will not bring back Andrei Kostitsyn, although the Predators later re-signed brother, Sergei. On 14 September 2012, Kostitsyn returned to Russia in signing an initial one-year contract with Traktor Chelyabinsk of the KHL.
In the 2014–15 season, his third with Traktor, Kostitsyn was traded to inaugural club, HC Sochi on 29 October 2014.
On June 1, 2017, Kostitsyn continued his career in the KHL, signing a one-year deal with Chinese outfit, HC Kunlun Red Star.
Kostitsyn represented Belarus extensively at the under-18 and -20 levels. He first played internationally at the 2000 World U18 Championships as a fifteen-year-old, but Belarus would struggle in the top division, finishing last. The next year, Kostitsyn competed with Belarus' under-20 team in the 2001 World Junior Championships' top division. Later that year, having been relegated for the 2001 World U18 Championships, Kostitsyn returned to record a tournament leading 14 points in five games within the second-tier to win the Division I championship.
Kostitsyn made his second under-20 appearance at the 2002 World Junior Championships and scored 3 goals to help Belarus stay in the top division. Later that year, Kostitsyn returned to the top division at the 2002 World U18 Championships, recording 10 points in 8 games and helping lead Belarus to a fifth-place finish. Competing in his third under-20 tournament at the 2003 World Junior Championships, Kostitsyn recorded two goals and one assist, but was relegated with Belarus to Division-I for 2004. Playing in the second-tier, he notched five goals and ten points in five games to help Belarus return to the top division in 2005, where he scored five points in his fifth and final World Juniors.
In 2004, Kostitsyn had also made his senior international debut as a nineteen-year-old at the World Championships in Division-I. He scored six points in five games. The following year in 2005, however, he did not record a point in six games playing in the top division. Following his rookie season in the American Hockey League (AHL) in 2006, he played in his third consecutive World Championships and recorded five points in six games. At the 2008 World Championships, Kostitsyn managed three points in five games.
On 23 December 2009, Kostitsyn was selected along with his younger brother Sergei, Mikhail Grabovski and Ruslan Salei as the only four current NHL players to represent Belarus in the 2010 Olympics.