Harry Kane had struck the bar with a header for Tottenham shortly after the break but Spurs, who beat league leaders Manchester City on Saturday, were unable to find a way past Burnley keeper Nick Pope.
Sean Dyche's side had enjoyed a 3-0 win at Brighton on Saturday and the Lancashire club look have to rediscovered the defensive solidity and determination that has ensured their survival in previous campaigns.
The home side had the best of the first half but were unable to create real openings against a Tottenham defence which looked far from comfortable.
Spurs boss Antonio Conte brought on Harry Winks at the break and the visitors made the stronger start to the second half with Kane flashing a header from a free kick against the bar.
Pope pulled off a fine reflex save to keep out a close-range effort from Ben Davies but Burnley were soon causing problems for the visitors.
Dejan Kulusevski had a chance for Spurs, drifting a shot wide of Pope's right-hand post after a counter-attack in the 70th minute.
But then Mee, who had been dominant at the back for Burnley, showed his value at the other end with his well-taken header sending a wet and windy Turf Moor into ecstatic celebrations.
Rodriguez should have made sure of the win, seven minutes from the end, but somehow poked over from four metres out and he surely had his heart in his mouth when Pope saved from substitute Steven Bergwijn in the final minutes.
The result leaves Spurs in eighth place on 39 points, seven behind fourth placed Manchester United but with two games in hand.