A visibly nervous Spurs side struggled at Selhurst Park, with Christian Eriksen's 30-yarder just over 10 minutes from time was the difference.
Crystal Palace didn't deserve to lose and their night got worse when Mamadou Sakho was stretchered off with a suspected knee injury.
In north London, another cagey game was separated by a Robert Huth own goal, as Nacho Monreal's wild effort deflected in three minutes from time to earn Arsenal a valuable 1-0 win over Leicester City.
EPL Table | Premier League standings after Gameweek 34
The Gunners move back into sixth place, within three points of Manchester United in fifth place.
The drama continues into Thursday night when United travel to the Etihad Stadium to face fierce rivals Manchester City - who are fourth in the table - in what could prove to be a straight shootout for Champions League football.
Further down the table, Middlesbrough beat Sunderland 1-0, their first win of 2017, to boost their faint hopes of survival.