Brazilian midfielder Denilson fired the Gunners in front after only five minutes as Arsene Wenger's men looked to leapfrog Manchester United and Chelsea, who do not play until Sunday.
Vermaelen was then deemed to have been the last man when he pulled down Guillermo Franco, and was shown a straight red card by referee Martin Atkinson.
However, Manuel Almunia saved Alessandro Diamanti's penalty and captain Cesc Fabregas - back from a injury lay-off - made it 2-0 with nine minutes left from the same spot following a handball by Matthew Upson.
The Gunners could, of course, fall back down to third place should both United and Chelsea win tomorrow, but after another brave display, Wenger's youngsters remain very much in the hunt for a first title since the 'Invincibles' campaign of 2003/2004.
Arsenal swept into the lead when Denilson won the ball from Valon Behrami on the edge of the West Ham box, before playing a neat one-two with Nicklas Bendtner.
The Brazilian needed no second invitation as he stepped into the penalty area before sidefooting a low shot into the bottom right corner.
The pair had also combined for Arsenal's stoppage-time winner at Hull, where Bendtner tapped in after a long-range effort from his team-mate.
The Dane then tried his luck from 20 yards, but was just off target.
West Ham were awarded a free-kick in a dangerous position 25 yards out when Sol Campbell bundled over Franco.
Diamanti's effort was well struck, but flew over.
The Hammers, who are hovering just above the relegation zone, produced a decent response to their early setback.
Junior Stanislas got away down the right, and his cross was almost met by Mido at the far post.
Fabregas and Bendtner combined to release Eboue into the right side of the box, and his angled drive was just too far ahead of Andrey Arshavin, sliding in at the far post.
Arsenal were reduced to 10 men a minute before the break when Vermaelen was, on the advice of the assistant, adjudged to have pulled down Franco as they chased a long ball into the box - and deemed the last man by Atkinson who, from 20 yards away, went to his back pocket for the red card.
Wenger was fuming - but will feel justice was done when Almunia made a brilliant save down to his left to keep out Diamanti's spot-kick.
Song dropped back into the centre of defence, as Arsenal regrouped for the second half with Bendtner deployed as a lone striker, which limited Arsenal's attacking options.
West Ham, though, failed to make the most of their extra possession.
Matthew Upson, the former Arsenal defender, was cautioned for tripping Eboue to concede a free-kick 25 yards out on the right.
Fabregas curled his effort around the wall, but wide.
Arsenal suddenly found another gear, lifted by the Emirates Stadium faithful as Samir Nasri flashed a low ball across the face of goal from the right.
West Ham were still a danger, though, as Diamanti's drive was deflected behind for a corner.
Campbell forced a good save from Robert Green when he met Fabregas' free-kick, before Nasri was replaced by Bacary Sagna as Eboue went onto the right wing, while West Ham sent striker Benni McCarthy on for Mido.
Arshavin's drive was well held by Green on what was now a skiddy surface, before Cole cut inside and drilled a low shot onto the outside of the far post.
Arsenal secured themselves a two-goal cushion and some much-needed breathing space after Upson's handball with nine minutes left, as Fabregas made no mistake from the penalty spot as the Gunners recorded a sixth successive league victory.
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