Arsenal lost further ground on the Premier League leaders after they were held to a goalless draw at West Ham.
A frustrating evening at the London Stadium saw the Gunners dominate for long spells but they were unable to break down a stubborn home side.
And they were inches from a damaging defeat after Hammers sub Javier Hernandez hit the crossbar with a minute remaining.
While a third game without a win was another blow to Arsenal's Champions League hopes, a point was another step in the right direction for West Ham following Saturday's win over Chelsea.
The hosts knew they would have to defend resolutely, as they had against the champions and during their narrow defeat by leaders Manchester City.
And Moyes has instilled a level of competency and organisation into a side who had at times descended into high farce under previous manager Slaven Bilic.
Arsenal were wasteful, though, with Mesut Ozil and Jack Wilshere squandering glorious chances in the second half.
Wilshere was handed a first Premier League start of the season by Arsene Wenger, along with striker Olivier Giroud.
The duo have been linked with January switches to West Ham in a bid to boost their World Cup hopes, although they may be mindful that the same move does not seem to be working out for Joe Hart at the moment.
In the early stages of a scrappy first half West Ham had the ball in the net, only for Marko Arnautovic to be flagged offside as he nodded in Arthur Masuaku's cross.
Alexis Sanchez swept a good chance over the crossbar after being neatly set up by Wilshere and Giroud, and Alex Iwobi was denied by the foot of a post.
Penalty shouts were exchanged, to no avail, at either end with Manuel Lanzini going down under Granit Xhaka's challenge and Giroud's overhead kick striking Aaron Cresswell's hand.
Moments after the restart Mark Noble's chest-down found Arnautovic in the area but, off balance, he hooked his shot onto the roof off the net.
Arsenal were starting to look more dangerous and Giroud headed narrowly wide before Ozil wasted his big chance, shanking his shot off target from all of 10 yards out after the ball ricocheted back to him.
Wilshere could have capped an impressive display with a goal but scooped his shot over before Adrian, still keeping Hart on the bench, beat away a curling Sanchez free kick.
Yet it was West Ham who sensed the chance for victory and it almost came when Hernandez suddenly found himself free in the area only to lift his shot against the crossbar, with the ball bouncing perilously close to the line.