Only after more than half an hour of extra time could Victor Hedman score a decisive 3-2 and thus 4-1 in matches.
It was a draw after the first period, it was a draw after the second, and the third - and the fourth period.
But with just under six minutes left of period five between Tampa Bay and Boston, Victor Hedman got the puck behind Jaroslav Halak.
The Swede's shot went between the legs of Boston defender Brandon Carlo, and when Tampa Bay's Patrick Maroon made sure to obscure Halak, the cage keeper could not prevent the important goal that took the Florida team forward and sent Boston home.
Hedman became the big hero, but the team can also thank a reborn goal thief.
After a goalless first period, it was Ondrej Palat who broke the deadlock of the match.
After going goalless from their first nine games in the playoff bubble in Toronto, the Czech has now scored in four straight games against Boston, and has been a decisive factor in the team's progress.
His 1-0 here was equalized by Boston's David Pastrnak.
Tampa Bay regained the lead in the third period, but Boston fought back another period and a half in the playoffs when David Krejci with just over two minutes left got in 2-2 and arranged the extra time where Tampa Bay was finally the strongest.
In the semifinals, the winner of the quarter-final series between the New York Islanders and Philadelphia awaits.
There, the Islanders lead 3-1 in matches.