Kevin Shattenkirk scored the winner seven minutes into the sudden-death extra period when he sent a snap shot through a screen on the powerplay to put the Lightning on the cusp of their first Stanley Cup since 2004.
Brayden Point had two goals for the Lightning while Yanni Gourde and Alex Killorn also scored.
Dallas got two goals from Joe Pavelski while John Klingberg, and Corey Perry added one each.
Game Five is scheduled for Saturday in Edmonton, which was selected as the site for the entire series to help limit travel and minimize COVID-19 risks.
The only team in history to come back from a 3-1 deficit in a final to hoist the Stanley Cup were the 1942 Toronto Maple Leafs, who won four straight games after losing the first three against the Detroit Red Wings. (Reporting by Frank Pingue in Toronto; Editing by Raju Gopalakrishnan)