Although Gauthier was originally sceptical of Palát's ability to play in the QMJHL, he was named to the team's roster and finished the season with 40 points in 59 games.

[5] Palát received early recognition from the league after recording six points over two games in mid-October to help lift the Voltigeurs to second place in the overall standings.

[5] Leading up to the 2011 NHL entry draft, the Tampa Bay Lightning had planned to invite Palát to a tryout as a free agent if he remained undrafted.

[15] After attending the Lightning's 2011 training camp, Palát signed a three-year, entry-level contract and joined their American Hockey League (AHL) affiliate, the Norfolk Admirals, for the 2011–12 season.

[17] In Game 4 of the Calder Cup Finals against the Toronto Marlies, Palát assisted on one of Mike Kostka's goals to lift the Admirals to their first AHL championship.

[21] Despite the location change of the team, Palát remained on a line with Johnson and Panik and began to improve on his rookie seasons points total.

[23] Later that month, Palát recorded his first professional hat-trick on 26 February against the Rochester Americans to help the Crunch continue their eight-game win streak.

[29] Due to the NHL's waiver rules, Palát was required to be assigned to the Crunch by 3pmEST on 3 April 2013 in order to play in the Calder Cup playoffs.

In their first game back together, Palát and Johnson tallied three points while Panik scored the game-winning overtime goal to lift the Crunch over the Portland Pirates.

[37] However, the "Top Gun Line" struggled in the Finals after losing Panik to an injury,[39] and the Crunch eventually fell to the Grand Rapids Griffins in six games.

[40] Palát participated in the Lightning's 2013 training camp and preseason games[41][42] before making their 2013–14 opening night roster alongside top gun linemates Tyler Johnson and Richard Pánik.

[43][44] The trio were reunited at the NHL level, where they continued their dominance, and were often referred to as "The Kid Line" instead of "Top Guns" by local media.

[48] However, following a leg injury to Steven Stamkos, head coach Jon Cooper broke up the trio and put Palát and Johnson with Martin St.

[53] Once Stamkos recovered from his broken leg and was named captain of the Lighting in earl-March, Johnson and Palát were split up for the first time in the season.

[54] Stamkos spoke highly of Palát after becoming his linemate in March: "You see the way he plays, the work ethic he has, the skill set he has, the confidence he's gaining each and every game.

"[55] Palát finished the season leading the Lightning in scoring with 23 goals and 36 assists for 59 points in 81 games, helping the Lighting qualify for the playoffs for the first time since 2011.

As such, Palát was named a finalist for the Calder Memorial Trophy, awarded to the NHL's rookie of the year, along with teammate Tyler Johnson.

[59] On 20 April 2014, Palát recorded his first career playoff goal and point against Carey Price in the Lightning's game three 3–2 loss to the Montreal Canadiens.

Due to various injuries throughout the lineup, head coach Jon Cooper placed the duo on a line with Nikita Kucherov for the second and third periods.

[66] Once Palát returned to the lineup, the "Triplets line" continued to collect points and helped the Lightning end the 2014 calendar year with four straight wins.

[68] While Palát had accumulated fewer overall points than his linemates, head coach Jon Cooper referred to him as "the straw that stirs the drink on that line" and called him a potential Selke Trophy winner.

[77] In their six game series with the Montreal Canadiens, the "Triplets line" combined for 17 points en route to the Eastern Conference finals for the fist time since 2011.

[126] Despite winning the Presidents' Trophy as the team with the best overall record during the regular season, the Lightning were unexpectedly swept in four games by the Blue Jackets in the first round.

[5] Due to their early exit from the playoffs, Palát was able to compete with the Czech Republic men's national team in the 2019 IIHF World Championship.

[134] Despite not scoring any goals in their first-round series, Palát and the Lightning defeated the Blue Jackets in five games to advance to the Eastern Conference semifinals against the Bruins.

[161] After defeating the Islanders in the third round for a second consecutive year this time in seven games, the Lightning faced the Montreal Canadiens in the 2021 Stanley Cup Finals.

[156] However, they failed to qualify for the Olympics after losing to Team Switzerland in the group stage and NHL players were unable to participate due to the spread of the omicron variant of COVID-19.

[170] Following his 10th season in the NHL with the Lightning, Palát left the club as a free agent and signed a five-year, $30 million contract with the New Jersey Devils.

[177] On 6 January 2014, Palát was named to the Czech Republic men's national ice hockey team to compete at the 2014 Winter Olympics in Sochi.

[181] Palát earned his first medal with the Czech Republic men's national ice hockey team at the 2024 IIHF World Championship.