Mikael Backlund scored twice, and Jacob Markstrom made 28 saves for the Flames (5-4-2), who have lost five in a row (0-3-2).
"We made some mistakes that cost us," Backlund said. "Maybe that happens when the team isn't feeling at their best, but we've got to turn this around. It starts with one win, and then we've got to go from there."
Lee cut Calgary's lead to 3-2 at 10:08 of the third when he scored on a rebound of Brock Nelson's initial shot, and Palmieri tied it 3-3 at 11:07 with a shot from the top of the left circle under Markstrom's left pad.
"Everyone was going in the third," Lee said. "We knew it was a game we had an opportunity to come back in. Once we got to our game, everyone that was going out there was making something happen."