The Gunners dominated for large periods at the London Stadium but they struggled to create clear-cut opportunities as they made it just two wins from nine away fixtures this season.
ArseneWenger opted to hand Jack Wilshere and Olivier Giroud their first league starts of the campaign, but it was Alexis Sanchez who had the first opportunity for the visitors, with his low strike being well saved by Adrian at the near post.
However, West Ham thought they had taken the lead on the quarter-hour mark when Marko Arnautovic powered a header into the top corner, only for the assistant to correctly rule that the Austrian international had been half a yard offside.
The first half was lacking in action, but Arsenal also came close to an opening goal in the 29th minute when Alex Iwobi switched the ball onto his left foot from around 10 yards out, but his attempt hit the post and rebounded away from danger.
West Ham had made it to half time without needing to get out of second gear in order to remain on level terms, but it was a different story after the restart.
Mesut Ozil was also presented with a chance inside the penalty area but, despite only having Adrian to beat, the German playmaker poked wide from 12 yards out.
As the game moved into the final quarter, a knockdown from Giroud gave Wilshere the chance to get on the scoresheet from 10 yards but, under pressure from a defender, the midfielder blazed high over the bar.
Sanchez was the next Arsenal player to threaten an opener, but the Chilean's 30-yard strike from a set piece was comfortably dealt with by Adrian, who was able to punch the ball clear of danger.
West Ham soon began to settle for a point with only Arnautovic left to contest balls against the Arsenal defence, and it worked for Moyes who continued to watch on as his players threw themselves in the way of each of their opponent's attacks.
Despite having to cope with incessant pressure, West Ham almost won it in the final minute of normal time when Javier Hernandez hit the underside of the crossbar after the ball had fallen to him inside the penalty area.
A goal would have been harsh on Arsenal, but Wenger will take little away from another game with dropped points on the road, while Moyes will have mixed feelings about the contest ending in a stalemate.
WEST HAM UNITED (3-4-3): Adrian; Reid, Ogbonna, Cresswell; Zabaleta, Noble, Obiang, Masuaku; Lanzini, Arnautovic, Antonio (Hernandez 82')
ARSENAL XI (4-3-3) Cech; Bellerin, Koscielny, Monreal, Maitland-Niles; Xhaka, Wilshere, Iwobi (Welbeck 70'); Ozil, Sanchez (Lacazette 82'), Giroud