The hosts started brightly and full of intent as Nikica Jelavic forced Wojciech Szczesny into a smart low save before Podolski called Steve Harper into action at the other end to palm away his strike.
However, it was Arsene Wenger's side who took the lead on 31 minutes as Ramsey netted his first goal since returning from injury, latching onto Santi Cazorla's pass inside the area to fire low past Harper.
Hull came mightily close to equalising within minutes as Jake Livermore smashed the post, before Podolski added Arsenal's second with a powerful strike into the bottom corner.
Podolski then added his second and the Gunners' third of the afternoon from close range after Harper had diverted Ramsey's effort into his path.
The game was preceded by a minute's silence in memory of former City full-back Andy 'Jock' Davidson, whose 579 appearances represent a club record.
Visiting keeper Szczesny foiled Hull captain Curtis Davies from a corner in the opening stages as the hosts started brightly.
Play was then bizarrely interrupted when a section of pitchside advertising hoardings blew over, littering the pitch but fortunately without injury to any of the players.
Mesut Ozil was denied a penalty when he tumbled over Ahmed Elmohamady's challenge, with replays indicating some contact.
Jelavic's sharp shot on the turn then drew an impressive low save from Szczesny but Shane Long took too long to line up his shot from Livermore's sumptuous through-ball, firing high and wide.
In between, Harper beat away a decent strike from Podolski's weaker right foot.
Elmohamady broke up a promising Gunners attack by poking the ball away from Podolski - but the visitors took the lead in the 31st minute.
A slick passing move ended with Ozil and Cazorla playing in Ramsey, who had been involved early in the move back at halfway, to fire low inside Harper's far post.
An Arsenal win, perversely, could help Hull's Europa League claims - a top-four finish for the Gunners would see the cup-winner's place revert to Hull no matter the result on May 17.
But on-loan Spurs man Livermore was in no mood to settle for the status quo, and sent a low drive against the inside of a post with Szczesny hopelessly beaten.
Long failed to make a meaningful contact with Liam Rosenior's cross and was punished by a second Arsenal goal on the stroke of half-time.
Hull were incensed as Jelavic was not awarded a free-kick after Arteta's challenge - a borderline call - and when Arsenal countered quickly, Ramsey chested Giroud's cross down to Podolski who rifled home.
Cup hero Matty Fryatt replaced Long at half-time but could not get the better of Bacary Sagna and Szczesny from his first opportunity.
Nine minutes into the second half, Podolski made it three. Harper saved well with his foot from Ramsey's initial effort but could only present the German with a simple finish.
Jelavic's shot was deflected away for a corner, and from it the Croatian's header forced another, but Hull could not find the cutting edge.
Harper had a real let-off when he misread Rosenior's back-pass and hacked the ball against Giroud. The France striker chased down the loose ball but hit the bar from an acute angle.
Rosenior's cross eluded substitute Stephen Quinn, newly on for Jelavic, and at the other end Per Mertesacker almost diverted Cazorla's poor free-kick on goal.
Harper saved well from Ramsey before an increasingly rare Hull threat saw Tom Huddlestone's shot tipped over by an unsighted Szczesny.
Fryatt stung Szczesny's palms from range but the match fizzled out in the closing stages, with Arsenal content with their winning margin.