Cheikhou Kouyate, Håvard Nordtveit and James Collins went close as West Ham remained resolute at the back to silence Romelu Lukaku and edge ever closer to securing their top flight status.
Everton were largely frustrated in the opening period, with the Premier League's leading scorer Lukaku starved of service.
The visitors had plenty of possession, but couldn't work the ball into threatening areas as Collins inspired a steelly defensive performance from the hosts.
And the West Ham could have opened the scoring after Nordtveit and Kouyate had tried their luck with unconvincing efforts from range.
The West Ham defender climbed at the back post to head a Lanzini cross wide of the Everton post.
Ronald Koeman threw on Ademola Lookman and Gareth Barry after the break and the former intially added some sparkle to the Everton forward play.
He twice threatended down the right before twisting past Winston Reid in the West Ham box and freeing Ross Barkley, but it came to nothing.
And West Ham enjoyed the better of the play thereafter.
Lanzini drilled an effort into Phil Jagielka's face after Edimilson Fernandes had somewhat fortunately worked the ball to his team-mate in the Everton box.
And the Argentine was again involved as he and Andre Ayew combineed beautifully, before Lanzini's Rabona cross was whipped into a dangerous area.
Lookman and Everton were reduced to unconvincing efforts from range as the West Ham defence blocked all routes to Lukaku.