Tomas Tatar scored for the Red Wings, who have lost 11 straight regular-season games to the Lightning. Jimmy Howard made 28 saves.
Detroit (23-24-9, 55 points) trails the Carolina Hurricanes by eight points for the second wild card into the Stanley Cup Playoffs from the Eastern Conference.
"I thought we've been closer, we had two wins and we lose this one," Red Wings captain Henrik Zetterberg said. "We're close, we just have to keep getting points and keep winning games. We did a lot of good things here today, but there are things we have to be better at."
Stamkos gave the Lightning a 1-0 lead 3:55 into the game when he scored with a one-timer from the slot off a pass from Hedman.
Kucherov scored on the power play at 7:05 of the first period to give the Lightning a 2-0 lead.
"They are good at taking care of their opportunities," Red Wings defenseman Jonathan Ericsson said. "We had our chances tonight too, but we didn't capitalize on them. From time to time, we played really good, but we don't capitalize and they do."