Nazem Kadri scored for the Flames (7-7-2), who opened a six-game road trip. Jacob Markstrom made 20 saves.
Calgary is 2-7-1 in its past 10 games.
"We were a little sluggish in the first period, 'Marky' made two or three really good saves," Flames coach Darryl Sutter said. "I think we probably had enough opportunities to at least tie going into the third."
After Flames forward Milan Lucic had a goal disallowed for offside at 7:28 of the first period, Stamkos gave the Lightning a 1-0 lead at 8:03 with a shot from the high slot.
Kucherov made it 2-0 with a power-play goal at 18:16 of the second period when he one-timed a feed from Stamkos in the right face-off circle.
"The power play has been an area that we wanted to improve on, and I think we have," Stamkos said. "We're too skilled of a unit to not at least get chances. Obviously, 'Kuch' is a big part of that."