Of Slovak lineage, Stastny is the son of Peter Šťastný, a Hockey Hall of Famer who played for the Colorado Avalanche's predecessor, the Quebec Nordiques, and finished his career with the St. Louis Blues.

As a dual citizen, Stastny has chosen to represent the U.S. in international hockey competitions, including the 2004 Viking Cup, the 2007 IIHF World Championship, and the 2010 and 2014 Winter Olympics.

He has mentioned "religion, education and the importance of family" as principle values in his upbringing,[8] and has spoken about his father's help in making him a better player.

[11] Their brother Marián arrived a year later, after Peter and Anton had raised the US$30,000 needed to bribe Czechoslovak government officials.

[13] Stastny began his junior ice hockey career in 2002 with the USHL's River City Lancers, playing with the team for two seasons and scoring 107 points in 113 games.

[8] Despite entering college hockey at a younger age than usual for a USHL player, Stastny scored 45 points in 42 games in his first season with the Pioneers, helping them win the MacNaughton Cup and Broadmoor Trophy.

[18] He then helped the team win its second NCAA Men's Ice Hockey Championship in a row by scoring two power-play goals in the final game at the 2005 Frozen Four tournament against the North Dakota Fighting Hawks.

[2][18] However, Steve Konowalchuk's career-ending heart problem opened a roster spot and Stastny's play impressed Avalanche coach Joel Quenneville.

[38][40] At the start of the season, Wolski was the Avalanche player seen as favorite to contend for the Calder Memorial Trophy; however,[41] the scoring streak put Stastny into contention as well.

[42] Stastny's play was one of the reasons the Avalanche experienced their best run of the season towards the end,[43] winning 15 of their last 19 games but missing the playoffs by one point.

Stastny ended his rookie season with 78 points, finished second to Pittsburgh Penguins' Evgeni Malkin in the voting for the Calder Memorial Trophy and was named to the 2006–07 NHL All-Rookie Team.

[57] Colorado ended the series by winning the sixth game and progressed to meet the Detroit Red Wings in the Western Conference Semifinals.

[1] Stastny ended the injury-plagued 2008–09 season with 11 goals and 25 assists for 36 points in 45 games while the Avalanche struggled heavily as a team and finishing last in the Western Conference.

He stayed uninjured the entire season, and only missed one game as a healthy scratch after the Avalanche narrowly clinched a playoff spot the previous evening as the eighth and final seed in the West.

Stastny's second career appearance in the post season ended after the top-seeded San Jose Sharks eliminated the Avs in six games in the first round in the 2010 playoffs.

With the delay of the 2012–13 season due to the lockout, Stastny followed his brother's footsteps to Germany and signed his first European contract with EHC München of the Deutsche Eishockey Liga on November 15, 2012.

[66] Stastny appeared in 13 games for Red Bull climbing to third among the team with 18 points before returning to the Avalanche upon the tentative lockout resolution on January 6, 2013.

[69] Upon signing Stastny, Blues' general manager Doug Armstrong said he envisioned using their new player as one of the "top-two centers to compete in the deep Western Conference.

[1] During their series against the Dallas Stars, Stastny played on the Blues' top line with Troy Brouwer and Alexander Steen.

[1] In the 2019 Stanley Cup playoffs, he found chemistry centering a line with Mark Stone and Max Pacioretty, scoring eight points in seven games.

[1] In Stastny's final campaign with the Golden Knights, in the shortened 2019–20 season, he began the year centering the team's second line with Reilly Smith and Jonathan Marchessault.

His offensive output declined during the season, which he hinted in December was due to a lingering injury, and he was subsequently relegated to the team's third line alongside rookie Cody Glass and Alex Tuch.

[1] On October 9, 2020, with a year remaining on his contract, Stastny was traded by the Golden Knights back to the Winnipeg Jets in exchange for Carl Dahlström and a 2022 fourth-round pick.

Unable to help the Jets qualify for the playoffs, Stastny finished the season ranked sixth in team scoring with 21 goals and 24 assists for 45 points in 71 games.

[96] As a free agent from the Jets, Stastny joined his fifth club and first Eastern Conference team in signing a one-year, $1.5 million contract with the Carolina Hurricanes on August 23, 2022.

[102] Stastny was selected to play for the U.S. men's national ice hockey team for the first time in the 2007 IIHF World Championship.

[107] Stastny was again selected to play for the U.S. ice hockey team in the 2010 Winter Olympics in Vancouver where he won a silver medal.

Upon completion of a disappointing 2012–13 season with the Avalanche, he accepted an invitation to the 2013 World Championship event in Finland/Sweden and was selected as Team USA's captain.

[109] He finished the tournament in second place in individual scoring with 15 points in 10 games, resulting in selection to the World Championship All-Star Team.

[112] George Gwozdecky, Stastny's coach at the University of Denver, has complimented his intelligence, and his ability to pass and to "see" the ice.