Ange Postecoglou’s side took an early lead through Brennan Johnson at the rain-lashed London Stadium.
But Zouma quickly hit back to leave Tottenham with just one win in their last three league games.
Tottenham remain in fifth place, two points behind fourth-placed Aston Villa, who face title-chasing Manchester City at the Etihad Stadium on Wednesday.
If Villa beat City for the second time this season, they would move five points clear of Tottenham in the race to qualify for next season’s Champions League.
A Villa defeat would allow Tottenham to move above them into fourth place if they win their game in hand.
“It’s what you expect when you come here. They are a strong team and they sit compact, so you have to be patient,” Postecoglou said.
“Disappointed not to get a better outcome but I thought we handled it okay.
“I will be very surprised if all games aren’t tight. Everyone is fighting for something. We have to maintain our levels.”
West Ham are now without a win in their last four league games, but boss David Moyes could take heart from the way his side bounced back with a spirited display after last weekend’s meltdown.
The Hammers blew a 3-1 lead against Newcastle last weekend as three late goals condemned them to a 4-3 defeat.
Tottenham, without a win at West Ham since 2019, came from behind to beat Luton 2-1 on Saturday thanks to Son Heung-min’s late winner.
The momentum from that gritty success was still fuelling Tottenham as they snatched the lead in the fifth minute in east London.
West Ham’s Jarrod Bowen scuffed wide from close range in the opening seconds and the visitors immediately made him pay.
Timo Werner was the catalyst for Tottenham’s opener with an astute cutback that Johnson converted with a clinical finish from five yards for his fifth goal this season.
Tottenham needed to win by at least two goals to climb above Villa and Pedro Porro nearly doubled their advantage with a long-range strike that fizzed just wide.
Confusion between West Ham keeper Lukasz Fabianski and Vladimir Coufal ended with the ball running to Son, but Tottenham’s captain couldn’t apply the finishing touch to punish that slack defending.
West Ham made the most of those escapes to equalise against the run of play in the 19th minute as Zouma rose highest to meet Bowen’s corner.
The French defender misjudged his attempted header but still managed to deflect the ball into the far corner off his back.
Bowen weaved through the suddenly anxious Tottenham defence for a deflected shot that Guglielmo Vicario saved at full stretch.
Vicario made an even better stop to push James Ward-Prowse’s dipping free-kick over the bar.
West Ham were on top again after the interval and Michail Antonio’s close-range effort was repelled by Vicario before Lucas Paqueta curled narrowly wide from 25 yards.
As the pressure mounted on Tottenham, Antonio held off Micky van de Ven but hit his shot straight at Vicario with just the keeper to beat.
After labouring for much of the half, Postecoglou’s men almost snatched the points in stoppage time, but Destiny Udogie’s shot was too close to Fabianski.
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