Mathieu Joseph scored a short-handed goal on the 5-on-3 for the Senators (4-5-0), whio have lost three straight. Anton Forsberg made 30 saves.
"We made some really bad plays in the third period that ended up in the back of our net," Ottawa coach D.J. Smith said. "You've got to learn to close games out or get the point when you have to, and we just didn't get it done. The penalty kill was outstanding, but 5-on-5 we had to keep grinding them down."
Dylan Gambrell gave the Senators a 1-0 lead at 5:44 of the first period. He recovered a loose puck in front of the net after the Lightning failed on multiple clearing attempts.
Sergachev tied it 1-1 at 3:17 of the second with a power-play goal, a one-timer from inside the point off a pass from Stamkos.
"The mental toughness of this team is on another level," Sergachev said. "But we've still got a long way to go. I don't think we're playing that playoff hockey that we're used to."
Joseph put Ottawa back ahead 2-1 at 12:21 after he beat Stamkos to a loose puck at center ice and scored on a breakaway.
"We have to be a little bit more mature," Joseph said. "Being [ahead] 2-1 after two periods, we didn't feel like we played bad, we just couldn't get anything going with all of those penalties."