Son Heung-min's contentious first-half equaliser looked to have kept Spurs at the summit on goal difference after Mohamed Salah's 14th goal of the season put the champions in front.
Firmino rose to head home Andy Robertson's corner in the 90th minute to dramatically win it, though, and put the Reds three points clear of Jose Mourinho's side.
Spurs felt Firmino's goal should have been ruled out for a foul by Jordan Henderson on Eric Dier when Robertson's delivery came in, but Spurs' 11-match unbeaten run in the top flight came to a halt.
