Arsene Wenger's men were the better side for most of the match and pushed West Ham's defence to their limits at times, but in the end they were not clinical enough to take all three points, leaving them in seventh behind Burnley.
The Gunners did not fall into the trap which snared Chelsea at London Stadium on Saturday, avoiding an early concession and swiftly establishing themselves as the superior force.
Chances were by no means a regular occurrence, though, with Alex Iwobi's 27th-minute effort which hit the post the only clear-cut opportunity of the first half.
Arsenal became even more dominant after the break and they looked more threatening in attack as a result, but they remained wasteful, with Olivier Giroud and Jack Wilshere – who was otherwise lively – particularly culpable.
Javier Hernandez almost paid them pay for their profligacy when he struck the crossbar a minute from time, but Arsenal survived that scare to leave with a point.
A clear pattern of play was established early on as the visitors got plenty of players forward and enjoyed periods of significant pressure.
That approach brought about the first chance nine minutes in, as Arsenal patiently passed around the edge of the area until Alexis Sanchez took up possession, cutting on to his right foot and forcing Adrian into a smart save down to his right.
West Ham gradually grew into the game and looked to have taken the lead in the 14th minute, but Marko Arnautovic strayed offside before heading in an Arthur Masuaku cross at the end of a rapid counter.
Arsenal went mightily close to the breakthrough themselves just before the half-hour mark, with Iwobi smashing against the right-hand upright from 12 yards after brilliant link-up play from Mesut Ozil and Sanchez.
The hosts made a strong start to the second half and crafted an early chance, as Mark Noble chested the ball over the Arsenal defence for Arnautovic, but the Austrian sliced his volley onto the roof of the goal under pressure from Nacho Monreal.
Arsenal soon began to reassert their authority, however, and Adrian was forced into action, scuttling quickly across his line to keep out Granit Xhaka's long-range effort.
But Adrian could only watch in the 55th minute when Giroud – making his first Premier League start of the season – headed just wide of the target.
West Ham had another lucky escape 20 minutes from the end, as Wilshere smashed over from seven yards after Giroud nodded a cross down to him.
Wenger's side piled the pressure on towards the end, but Sanchez's 30-yard free-kick – which Adrian pushed away from the top-right corner – was as close as they went.
West Ham then almost stole all three points right at the end, but Hernandez fired an effort onto the crossbar from 12 yards.
And that proved to be the final chance of the match, as West Ham built on Saturday's 1-0 defeat of Chelsea with another commendable result.
Key Opta facts:
- Only once in Premier League history have Arsenal won fewer than nine points from their first nine away games of a season (eight in 2005-06).
- Eight different clubs have won more points away from home in the Premier League than Arsenal (9) this season - Manchester City, Chelsea, Manchester Utd, Liverpool, Burnley, Watford, Leicester City and Tottenham.
- West Ham failed to have a single shot on target in a Premier League home match for the first time since August 2013, in a 0-1 defeat against Stoke City.
- The Gunners are now winless in three consecutive Premier League games for the first time since April.
- Arsenal have kept a clean sheet in two of their last three Premier League away games, as many as they had kept in their previous 14 on the road.