Brock Nelson and Parker Kelly scored for the NHL-leading Avalanche (31-4-7), who have lost consecutive games in regulation for the first time this season. Scott Wedgewood made 24 saves.
Colorado finished 1-2-0 on a three-game road trip, defeating the Carolina Hurricanes 5-3 on Saturday before losing to the Florida Panthers 2-1 on Sunday.
"Tough road trip, no doubt about it. Three good teams," Wedgewood said. "Obviously, we want as many points as possible, but I think either one of those (losses) coming down to the wire you could've stolen two or four more."
Hagel put Tampa Bay ahead 3-2 at 8:31 of the third period, scoring with a one-timer from the right face-off circle to the short side off a backhand pass from Max Crozier.
"Those are fun games to play in and obviously you can sit and probably say any team could've won, but it came down to the next goal and we were able to find it," Hagel said. "They've got an incredible team over there, but we've got an incredible team here."
Jake Guentzel gave the Lightning a 1-0 lead by scoring a power-play goal with 12 seconds left in the first period. Kucherov found him open below the right face-off circle, and he flipped the puck over Wedgewood's stick.