Spurs looked destined for a frustrating afternoon at White Hart Lane as summer signing Vincent Janssen and Harry Kane were both thwarted by a resolute Palace back line.
Instead Wanyama, bought from Southampton in June, capped his home debut with an 83rd-minute winner, heading in from close range to snatch a dramatic 1-0 win.
Tottenham fans arriving for their first game of the season would have been surprised to see their ground with a gaping hole in the north east corner, as the construction of the club's new stadium has meant the removal of 4,000 seats at one end.
They may also have looked twice at the teamsheet as Pochettino dropped Dele Alli and instead rewarded Janssen, impressive off the bench against Everton last weekend after joining from AZ Alkmaar, with a starting spot up front.
Palace had announced the signing of their own blockbuster forward, Christian Benteke, less than five hours before kick-off, but the Belgian was not involved in the squad and will have to wait until Tuesday's EFL Cup game against Blackpool for a chance to make his debut.
Hugo Lloris' hamstring injury meant Michel Vorm started only his fifth Premier League fixture since joining Spurs from Swansea two years ago and the Dutch shot-stopper was drawn into the action almost immediately as Jason Puncheon tested him with an early shot.
It was Tottenham who dominated the opening stages, however, as Toby Alderweireld nodded a free header straight at Wayne Hennessey early on and Christian Eriksen, a pale shadow of his dynamic self in the opening half, botched a counter-attack by tripping over the ball when Spurs had four against two.
Janssen could have scored after Hennessey parried Kane's deflected effort from distance but despite twice trying to steer home the rebound, the Palace goalkeeper stood tall and made the saves
Kane twice went close either side of half-time, first with a driven shot just wide and then with a glancing header under pressure from Pape Souare, but as the home fans became impatient, Palace grew in confidence.
Yohan Cabaye, a second-half substitute, should have given the visitors the lead on the hour-mark but skied over the crossbar and that was enough warning for Pochettino to introduce Alli, who replaced the out-of-sorts Eriksen.
Alli's impact was instant, his bending pass putting Janssen through one-on-one, only for the striker to skew his finish wide with his weaker right foot.
Palace supporters were eager to remind Alli of his disappointing Euro 2016, chanting "you let your country down" whenever he was involved, but the midfielder was unperturbed, instead firing a shot just over the crossbar late on.
Spurs looked to be running out of ideas, only for Wanyama to transform the mood completely as Kane headed Erik Lamela's corner towards goal and the Kenyan was perfectly placed to nod into the back of the net and seal a hard-fought three points.
Source : PA
Source: PA