West Ham moved one step closer to safety and Everton moved above Arsenal into sixth place after the two sides played out a 0-0 stalemate at the London Stadium.
Neither side was able to find a way to break the deadlock with Everton striker Romelu Lukaku failing to extend his run of having scored in his nine previous games against the Hammers.
The Premier League leading scorer was not the only player to struggle during a drab game that left both managers frustrated.
Victory for West Ham would have taken them to the 40-point mark while Everton would have moved above Manchester United and into fifth place with a win.
West Ham blew the chance to effectively secure their Premier League future seven days previously when they twice allowed bottom club Sunderland to equalise before claiming a point in a 2-2 draw.
That was the latest disappointment in a dismal run for Slaven Bilic's side and the manager responded by dropping keeper Darren Randolph, at fault for both Sunderland goals, in favour of Adrian who made his first league appearance since early November.
A lengthy injury list together with suspensions for Mark Noble and Sam Byram meant Bilic was forced into other changes, including drafting in Jonathan Calleri for his first league start since arriving on loan from Deportivo Maldonado last August.
It came as no surprise then, that the home side appeared disjointed and struggled to make an impression on the well-organised visitors.
Adrian in particular looked ring-rusty when he took too long to clear after Edimilson Fernandes directed a throw-in towards his keeper. The Spaniard's delay allowed Lukaku to block the attempted clearance and Adrian was fortunate to gather the loose ball comfortably.
Lukaku remained isolated up front and starved of quality service with Calleri and Andre Ayew made little impression on the Everton back-line.
There were more promising signs for Bilic's side, however, during the latter stages of the first half when they twice came close to opening the scoring after the visitors defence were unsettled by crosses from either flank.
West Ham left back Arthur Masuaku's delivery was excellent and forced a corner that led to Manuel Lanzini picking out James Collins at the far post where the centre-back directed a header wide.
Then four minutes before the break, Everton were again forced into some desperate defending after a goalmouth scramble that followed Lanzini's free-kick.
Koeman was understandably dissatisfied with his side's first half efforts and reacted by replacing Tom Davies and Idrissa Gueye with Gareth Barry and Ademola Lookman in an effort to lift the side.
The opening period of the second half, though, maintained the pattern of the first with West Ham continuing to press with Lanzini unfortunate to see a powerful shot deflected clear after striking Everton defender Phil Jagielka in the face.
Lukaku had remained a peripheral figure, never looking like adding to his 24 league goals, but finally burst into life with a positive run that was stopped by Collins' challenge.
And Lookman's presence had the desired effect from Koeman's point of view with the teenager producing Everton's first shot of the game in the 70th minute.
The winger's long-range effort flew wide, but he quickly followed that up with a rising shot struck from the edge of the penalty area that almost tested Adrian.
That, though was as close as Everton came to breaking the deadlock with West Ham also producing little more of note as the game petered out into a draw.