His decision to drop Joel Robles following noticeably poor form in recent weeks allowed Maarten Stekelenburg to make a first appearance since mid-December.
West Ham made four changes to the side that drew at Sunderland with Adrian, Winston Reid, Havard Nordtveit and Jonathan Calleri - who made his first Premier League start for the club - coming in for Darren Randolph, suspended Sam Byram, injured Andy Carroll and Robert Snodgrass.
The result sees Ronald Koeman's side remain seventh in the table while the Hammers climb to 13th.
Adrian in particular looked ring-rusty when he took too long to clear after Edimilson Fernandes directed a throw-in towards his keeper.
"During the season we've scored enough goals", Bilic said. But he barely had a kick as he was starved of service and could not carve out any opportunities himself.
He has scored in Everton's last nine home matches, the first player to do that since Dixie Dean in 1934.
West Ham have only won seven of the last 48 league meetings in the 30 years, losing 29 of those games.
In the 36th minute, another succession of set plays nearly ended in the opening goal after Winston Reid won the ball from a Manuel Lanzini corner, played the ball back to the Argentinian, whose second cross was met by James Collins at the back post, but his header was wide.
West Ham's interest in having Simone Inzaghi as their boss for the next season was taken with some surprise in Italy. Substitutes: Gareth Barry (for Davies, 45 minutes) 5, Ademola Lookman (for Gueye, 45 minutes) 6, Dominic Calvert-Lewin (for Mirallas, 71 minutes) 6.
It's clear that Barkley's influence makes the team tick, but the onus can not rest exclusively on the England global exceeding realistic expectations of him.
Everton's Dutchman totally misjudged Manuel Lanzini's inswinging corner and was lucky it went between his post and the onrushing Cheikhou Kouyate. Lookman collects the ball 40 yards out, does a neat half turn and lashes a shot wide of goal from distance.
One of the crosses into the box saw Ashley Williams denied a chance to head goalwards by compatriot Collins, who pulled back on his shirt but avoided any punishment from referee Roger East.
The Spaniard's journey from backbench to frontline came full circle with last weekend's rashly conceded penalty against Burnley potentially ending his season.