Ashley Barnes put the visitors ahead in the 21st minute but the lead lasted only six minutes, Dele Alli equalising from a cross by Kyle Walker.
Another England international, fullback Rose, hit a fierce winning goal from a pass by Moussa Sissoko, who could have been sent off earlier for a challenge on Stephen Ward.
Tottenham stay fifth, 10 points behind leaders Chelsea, while Burnley remain three points above the relegation places.
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Although Welsh international Sam Vokes was close to converting a cross from Andre Gray in added time, the result confirmed the Lancashire side's poor away record.
They have now taken one point from eight games on the road and Barnes's goal was only their second away from home.
The visitors felt aggrieved that Sissoko was still on the field to make the winning goal.
He had replaced Mousa Dembele in the 62nd minute, caught Ward on the knee with a high tackle seven minutes later and then immediately sent Rose through to score.
"I don’t want go on about the referee but Moussa Sissoko has to go off and he sets up the second goal," Burnley manager Sean Dyche said.
"Obviously I am very disappointed about that. They are a top-class side but we do need the rub of the green.
"My player's reaction to a clear sending off is that he gets up. If he does 14 rolls then the ref is under pressure. It is a clear sending off in my opinion.
"It was a blind challenge, a high challenge and simply a red card."
Tottenham manager Mauricio Pochettino declined to be drawn into the controversy.
"The referee saw it better than me," he said.
"The team played in a very good way and we fully deserved the victory. I am pleased for that. It was a very difficult game."
(Reporting by Steve Tongue; Editing by Ken Ferris and Clare Fallon)
