Spurs could have narrowed the gap to a point with victory at London Stadium, but were well below par as the Hammers helped Chelsea close on the title.
Hammers keeper Adrian made first-half saves from Harry Kane and Eric Dier before Lanzini smashed in a loose ball.
Chelsea need two wins from their final four games to be crowned champions.
Tottenham must hope the Blues slip up in a favourable-looking run-in, which includes home games against three sides in the bottom seven.
Antonio Conte’s side will secure the title on Friday, 12 May if they beat both Middlesbrough and West Brom.
