On 10 October 2011, the Lightning signed Palát to a three-year, entry-level contract. He then began his professional career with the Norfolk Admirals, Tampa Bay's American Hockey League affiliate in the 2011–12 season. While a member of the Admirals, the club would set a professional hockey record by posting 28 consecutive wins. Norfolk would go on to capture the Calder Cup as AHL champions. 's opening day roster to begin the 2013–14. Palát played the next season in the AHL, 2012–13, with the Syracuse Crunch, Tampa Bay's new AHL affiliate. After a brief call-up to the NHL, he scored his first career NHL goal on 16 March 2013, against Justin Peters of the Carolina Hurricanes. Palát would later be returned to Syracuse for the 2013 Calder Cup playoffs. After reaching the Calder Cup Final, Palát and his teammates had the unique opportunity at back-to-back AHL championships, despite playing for two different hockey clubs. The Crunch, however, would fall to the Grand Rapids Griffins in the Final. Palát would make the Lightning's 2013–14 opening roster out of training camp, along with his AHL line-mates Tyler Johnson and Richard Pánik. At the end of the season, he was nominated for the Calder Memorial Trophy, awarded to the NHL's rookie of the year, along with teammate Tyler Johnson. Palát, along with Johnson, became the first teammates to be nominated for the Trophy since Patrick Kane and Jonathan Toews of the Chicago Blackhawks in 2008. However, the Trophy eventually went to Nathan MacKinnon of the Colorado Avalanche, though Palát was named to the NHL All-Rookie Team. On 9 June 2014, the Lightning announced that they had signed Palát to a three-year contract extension. He had 23 goals, 59 points and a +32 plus-minus rating in 81 games for the Lightning in 2013–14. He ranked second among all NHL rookies in points, and was third in goals scored. Additionally, Palát became the third Lightning player to finish a season with a plus-minus rating higher than +30. On 14 July 2017, the Lightning announced that they had signed Palát to a five-year contract with an average annual value of $5.3 million. On December 31, 2017, Palat recorded his 162nd career assist in a 5–0 win over the Columbus Blue Jackets at Nationwide Arena. The assist moved Palát past Chris Gratton for tenth-most assists in franchise history. On November 23, 2019, Palat recorded his 100th career NHL goal in a 6-2 Lightning win over the visitingAnaheim Ducks. On November 25, 2019, Palat recorded his 200th career NHL assist and 300th career NHL point in a 5-2 Lightning win over the visiting Buffalo Sabres. Palat was the 8th Lightning player to record 200 career assists and 10th Lightning player to record 300 career points.
