Ramsey and Eduardo put Gunners through
However, once the action got under way, the encounter turned out to be a decent cup tie.
West Ham, battling relegation from the top flight, took the lead in first-half stoppage time when Alessandro Diamanti beat the offside trap to slot past Gunners goalkeeper Lukasz Fabianski.
But with 12 minutes remaining the outstanding Aaron Ramsey levelled and Eduardo's looping header after 83 minutes completed the turnaround.
The Gunners have a crucial Premier League game against Bolton on Wednesday which could see them narrow the gap to leaders Chelsea to a point.
But their quest for a first trophy in five years - since they won the FA Cup in 2005 - remained alive on more than one front as they entered the fourth round draw, where they were pitted against Stoke.
Even though Wenger had made five changes from the 4-1 win over Portsmouth in midweek, it was certainly not an inexperienced Arsenal side, with the likes of Carlos Vela, Fran Merida and Luke Wilshere included along with veteran defender Mikael Silvestre, captain in the absence of injured Cesc Fabregas.
The Hammers - famous winners over the Gunners in the 1980 FA Cup final at Wembley - had come back from 2-0 down here to draw their league encounter earlier in the season, and started well, with Diamanti firing over after cutting in from the right.
Merida had a low, 25-yard strike comfortably saved by West Ham goalkeeper Robert Green after 12 minutes before Arsenal slowly built some pressure on the home defence.
Wales teenager Ramsey and Merida both had shots charged down before James Tomkins gave the ball away, Eduardo pounced and the Croatian forced Green to make a full-stretch save as his 20-yard effort looked set to creep into the bottom corner.
West Ham responded after Alex Song - off to the African Nations Cup with Cameroon tomorrow - bundled over Daprela to concede a dangerous free-kick, 25-yards out. Junior Stanislas was just off-target with a dipping effort over the bar.
There was action at both ends as first Diamanti's curling shot from the left was palmed away by Fabianski before Thomas Vermaelen lashed his first-time shot over after a corner.
Stanislas stole possession on the edge of the Arsenal box, but the ball just would not drop for the England Under-21 forward.
After 40 minutes, Mexican Carlos Vela got on the end of a long ball up field, but his angled shot was blocked by Green at the right-hand post.
The Hammers took the lead in first-half stoppage time when Diamanti beat the Arsenal offside trap to collect Valon Behrami's pass and slot in off the post.
The hosts resumed in positive fashion after the interval, with England Under-19 forward Frank Nouble, who was making his debut, firing wide from the edge of the area before Stanislas' 25-yard shot was palmed away by Fabianski.
Arsenal made a double change after 65 minutes, replacing Merida with Diaby and Wilshere for Nasri.
The changes seemed to galvanise the visitors, who had lacked any real penetration in the final third.
Green made a fantastic double save to deny first Diaby after he burst into the penalty area and then the follow-up effort from Song.
There was, however, little the England goalkeeper could do about Arsenal's equaliser with 12 minutes left, when Ramsey smashed a low shot into the far corner after collecting Vela's pass through a crowded penalty area.
Before the Hammers could mount a recovery, Arsenal had turned the tie around.
Vela was again the provider, sending over a cross from the left, which Eduardo met ahead of Tomkins to loop the ball over Green into the top corner.
Arsenal could have further extended their lead, but former Gunner Matthew Upson cleared Vela's shot off the line and Green saved from Ramsey.
