Johnson broke the deadlock in a game that looked destined for overtime with his eighth goal of the playoffs, taking a centering pass in front of the net and pushing it past Montreal goaltender Carey Price.
Lightning goalie Ben Bishop carried a shutout well into the third period but Montreal tied the game when right winger Brendan Gallagher, diving to the net, pushed a rebound under his pad to tie the game at 1-1 with 9:57 left.
Lightning center Alex Killorn had scored a first-period goal to give Tampa the early advantage.
With eight minutes left in the opening period, Killorn was set up with a strong entry pass from Lightning center Steven Stamkos, who picked up his sixth assist.
Tampa Bay, winners of five straight in the playoffs, can finish the series with a win at home on Thursday night.
The Canadiens had more scoring opportunities through two periods, outshooting the Lightning 8-6 in the opening period and holding them without a shot for the first 12 minutes of the second. (Editing by Jahmal Corner)