The 2-1 win keeps Spurs' Champions League hopes alive while the Clarets now face next season in the Championship.
Jacob Bruun Larsen had the Clarets ahead before Pedro Porro and then Micky van de Ven struck for Tottenham.
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Afterwards, Spurs boss Ange Postecoglou said: "We were dominant enough, their keeper was outstanding today and we had enough chances.
"It's always difficult when you lose four matches in a row and it is only natural there will be a bit of anxiety and a bit of stress there.
"For the most part I think we handled the game well and we finished the game strong.
"It should have been a more comfortable victory but the most important thing is we won the game."