Trailing 3-0 after the first period, the Lightning came roaring back to score five straight goals, including three in the middle frame. Purcell scored two of them only 63 seconds early in the period and Sean Bergenheim got the equalizer at 10:53.
Ryan Malone stole a pass at center ice 6 1/2 minutes into the third period and shoveled a backhand pass to Simon Gagne in the high slot. The Lightning winger put it past Tim Thomas to put the Bolts up 4-3.
Gagne added two assists and Mike Smith stopped all 21 shots he saw after replacing starting netminder Dwayne Roloson in the first period as Tampa Bay knotted the series.
Patrice Bergeron scored twice and Michael Ryder also found the net for the Bruins, who will try to take back the momentum in Monday's Game 5 in Boston.
Thomas finished with 32 saves.