Kucherov took a return pass from Brayden Point on a 2-on-1 rush and scored into an open net. The play came right after Andrei Vasilevskiy stopped Alex DeBrincat on a partial breakaway.
Conor Geekie scored his first goal of the season, and Jake Guentzel and Erik Cernak also scored for the Lightning (50-25-6), who have won two in a row after losing three straight. Vasilevskiy made 27 saves.
Patrick Kane became the 24th player in NHL history to reach 1,400 points by getting an assist for the Red Wings (41-30-10), who are 1-3-2 in their past six games and have already been eliminated from playoff contention. DeBrincat, David Perron and Marco Kasper scored, and Cam Talbot made 18 saves.
Perron gave the Red Wings a 1-0 lead at 5:31 of the first period when he scored into an open net off a pass from J.T. Compher. The sequence started after Vasilevskiy couldn’t glove down Ben Chiarot’s shot from the point.
Geekie tied it 1-1 at 17:05, scoring with a forehand that trickled past Talbot on a breakaway.
Cernak gave Tampa Bay a 2-1 lead at 15:53 of the second period. Talbot stopped Cernak’s initial redirection and Ryan McDonagh’s rebound chance in front. However, McDonagh collected his own rebound near the right post passed back to an open Cernak, who scored past the prone goaltender.
Guentzel extended the lead to 3-1 at 18:03 with a wrist shot from the left circle that beat Talbot to the far post. Kucherov set Guentzel up after his initial shot was deflected wide.
Kasper cut it to 3-2 at 2:56 of the third period. He scored with his backhand from the edge of the crease off a pass from James van Riemsdyk, who took control of a loose puck after an initial shot from Justin Faulk.
DeBrincat tied it 3-3 at 15:16. Kane stole the puck from Emil Lilleberg at the defensive blue line, skated in on a 2-on-0 rush, and passed to DeBrincat for the finish.
Kane, who has 508 goals and 892 assists in 1,368 career games, is the fourth active player to reach the 1,400-point milestone, joining Sidney Crosby and Evgeni Malkin of the Pittsburgh Penguins, and Alex Ovechkin of the Washington Capitals.