The Italian's sixth goal of the season had given the Hammers a surprise interval lead, before the deadly Gunners duo secured a trip to Stoke City in three weeks' time with two strikes in the final 12 minutes.
Fran Merida's low 20-yarder forced Robert Green into an awkward save and then Ramsey had two rasping shots blocked by Valon Behrami, before Eduardo saw the Hammers keeper tip his 18-yard grass-cutter aside for Arsenal's fifth corner inside the opening 20 minutes.
Following last Monday's defeat at Tottenham Hotspur, injury-ridden Gianfranco Zola had made four changes as he handed full debuts to both Frank Nouble and Fabio Daprela, while Luis Jimenez and Junior Stanislas also received call-ups in place of the injured Herita Ilunga (hamstring) and Scott Parker (hamstring), Guillermo Franco (suspended) and the rested Jack Collison.
And compared to the Gunners, the Italian's 17th-placed side looked to have little firepower as the rookie Nouble and on-loan Chilean Jimenez saw few scraps come their way in the opening exchanges.
But Stanislas did manage to send a 25-yard free-kick into the shivering Arsenal fans before the frozen Lukasz Fabianski was finally called into action, when he was forced to claw away Diamanti's angled curler.
At the other end, a breaking Carlos Vela saw his angled shot saved at the near post by Green, who looked to have kept it goalless at the break, but further upfield his team-mates had other ideas.
Indeed, in added time, Diamanti sent the East Enders amongst the disappointing 25,549 crowd into ecstasy, when he raced on to Behrami's inch-perfect, defence-splitting pass and steered the ball into the net via the base of the exposed Fabianski's left-hand post.
Having snared an interval lead against the run of play, Zola's men emerged for the restart hungry for more, as the ox-like Nouble fired just wide before Stanislas saw his 15-yard screamer palmed away.
At the other end, Jack Wilshere scooped high over before Wenger replaced him and Merida as Samir Nasri and Abou Diaby came on for the final 25 minutes of a tie that was now really beginning to warm-up.
Certainly, that double switch looked set to pay quick dividends, when Diaby burst to the edge of the six-yard box, but Green gallantly dived into his studs before recovering to tip Alex Song's follow-up behind, too.
With just a dozen minutes remaining, however, the reshuffled Arsenal drew themselves level, when Vela collected from Song and picked out Ramsey, whose ten-yard shot on the turn gave Green no chance.
Then, on 83 minutes, Eduardo secured the Gunners' place in the next round when he met Vela's left-wing cross with a header that looped beyond Green and under the keeper's right-hand angle.