Jurgen Klopp's side went in front through Sadio Mane at London Stadium, although the goal should have been disallowed for a clear offside in the build-up.
Antonio quickly levelled for the Hammers, with his fourth career goal against Liverpool leaving the league leader just three points clear of chasing City, which had defeated Arsenal 3-1 the previous day.
The result could have been even worse for the Reds, Declan Rice wasting a free header shortly before the interval, although substitute Divock Origi almost won it in added time.
Javier Hernandez wasted two good early chances as Liverpool started poorly, but they took the lead in controversial circumstances in the 22nd minute.
Brilliant skill from Adam Lallana found James Milner, who cut the ball back for Mane to finish well, but replays showed the makeshift right-back was offside when he received the pass.
West Ham equalised within six minutes, though. Felipe Anderson caught out the Liverpool defence with a smartly taken free-kick, sliding a low pass into the right channel for Antonio, who shrugged off Naby Keita and drilled a fine finish across Alisson and in off the post.
Rice and Issa Diop missed headed chances either side of half-time as Liverpool continued to defend set pieces poorly, while Mohamed Salah could not make the most of two second-half openings he created for himself.
Hammers captain Mark Noble almost snatched all three points, his 73rd-minute drive flying narrowly over Alisson's goal.
But Origi could have settled matters in the 94th minute, the Belgian - a late hero against Everton earlier in the campaign - firing straight at Lukasz Fabianski after Keita picked him out with a chipped pass, the flag staying down again when striker was in an offside position