After arriving from Russia to play for Drummondville of the Quebec Major Junior Hockey League in 2008-09, Kulikov made an instant impact. He led all defensemen with 62 points in 57 games and was named the league's Defensive Rookie of the Year, Rookie of the Year, Best Professional Prospect and Defenseman of the Year.

The Florida Panthers were impressed enough to select Kulikov in the first round (No. 14) of the 2009 NHL Draft. He made his NHL debut at the age of 18 on Oct. 2, 2009, against the Chicago Blackhawks in Helsinki.

Kulikov got his first NHL point Oct. 10, 2009, against the New Jersey Devils, beginning a three-game assist streak. Kulikov scored his first NHL goal Nov. 7, 2009, against the Washington Capitals and had his first two-goal game in the League on Nov. 27, 2009, against the Toronto Maple Leafs.

Kulikov had his first multi-assist game in the NHL on Jan. 19, 2011, against the Columbus Blue Jackets and got a point in four consecutive games from March 12-19, 2011. He had 24 points in 2011-12 despite missing 23 games due to a knee injury. Kulikov had a five-game assist streak from Oct. 31- Nov. 10, 2011, and made his Stanley Cup Playoff debut that season, getting his first postseason point in the NHL with an assist April 24, 2012, against the Devils.

In 2014-15 Kulikov got his 100th NHL point with an assist Oct. 17, 2014, against the Buffalo Sabres. He scored the first Stanley Cup Playoff goal of his career April 15, 2016, against the New York Islanders. Kulikov got a point in each of the first three games of that first-round series.

On June 25, 2016, the Sabres acquired Kulikov and a second-round pick in the 2016 NHL Draft from the Panthers for Mark Pysyk and a second- and third-round pick in 2016. He became an unrestricted free agent after the 2016-17 season and signed with the Winnipeg Jets on July 1, 2017.

Kulikov signed with the Devils on Oct. 22, 2020 and was traded to the Edmonton Oilers on April 12, 2021. He had two assists in 10 regular-season games with the Oilers and did not score a point in three playoff games. After one season with the Minnesota Wild, Kulikov was traded to the Anaheim Ducks on Aug. 31, 2022. He was then traded to the Pittsburgh Penguins on March 3, 2023 for Brock McGinn and a third-round pick in the 2024 NHL Draft.

He signed a one-year contract to return to the Panthers on July 1, 2023, and became the first skater in NHL history and the second player overall to win the Stanley Cup after playing for eight different teams (Curtis McElhinney for the 2019-20 Tampa Bay Lightning). He signed a four-year contract July 1, 2024, and had an assist in his 1,000th NHL game, a 3-0 win for the Panthers against the Los Angeles Kings at Amerant Bank Arena on Jan. 29, 2025. Kulikov also helped the Panthers repeat as champions that season.

Kulikov has represented Russia in international play, including first-place (2007) and second-place (2008) finishes with Russia at the IIHF World Under-18 Championship and a third-place finish at the IIHF 2009 World Junior Championship.