The Championship club were protecting a slender 1-0 lead from a fortnight ago in Suffolk, and defended resolutely to frustrate the Gunners for more than an hour.
However, Nicklas Bendtner finally ended their resistance with a fine individual goal before Laurent Koscielny headed in a second and captain Cesc Fabregas kept Arsene Wenger's men on course for a first trophy since 2005.
Nine thousand Ipswich fans had made the journey from Suffolk, perhaps more in hope than genuine expectation of a cup shock.
Arsenal, who named a strong side, made a positive start, and Van Persie - who missed the first leg at Portman Road, but had struck six goals in his last three appearances - fired a volley just over the angle from Bacary Sagna's cross.
Fabregas used his programme notes to hit back at claims from Roberto Martinez he had dived to win a penalty during the 3-0 Premier League win over Wigan here on Saturday.
After 10 minutes, the Spain World Cup winner burst into the Ipswich box and went down under an outstretched arm from Gareth McAuley, but referee Mark Halsey waved play on.
McAuley was then in the thick of the action at the other end, heading over a deep free-kick into the Arsenal danger zone.
There was then a stoppage after Sagna needed treatment following an accidental clash of heads with his keeper Wojciech Szczesny when trying to defend.
The France international was unable to continue with what looked like possibly a broken nose, so Emmanuel Eboue replaced him after 18 minutes.
A clever flick from Van Persie released Fabregas into the right side of the Ipswich area, but his low shot was saved at the near post by Marton Fulop.
Van Persie then went within inches of levelling the tie when his header crashed against the bar from Bendtner's right-wing cross.
Bendtner's shot was blocked after Fulop dropped a free-kick in the six-yard box as the Tractor Boys continued to defend resolutely.
Jack Wilshere chipped the ball through to Fabregas in the left side of the box, but after bringing it down superbly, the skipper could only drill a low shot across the face of goal - after which the travelling fans took great delight in chanting '1-0 to the rugby team' in reference to Fabregas' post-match comments following the first leg.
It was more of the same at the start of the second half, as Arsenal had plenty of possession, but were faced with trying to break down an organised resistance, Ipswich deploying two banks of four men behind the ball.
The pressure, though, was relentless and when the breakthrough finally came after 61 minutes, it was thanks to a moment of real quality from the much-criticised Bendtner.
A quick ball from Wilshere out to the left was brought down superbly by the Dane, who then cut inside with a deft backheel before curling a low shot into the bottom right corner.
Before Ipswich could recover, it was 2-0.
Fulop came to punch a left-wing corner from Arshavin, but got nowhere near the ball as Koscielny headed in from six yards.
Chants of 'Wembley' started to ring out around the stadium as the Gunners fans started to dream of an end to their six-season trophy drought.
There was a warning, though, when substitute Jason Scotland broke into the left of the penalty area and his angled shot was saved at the near post.
Fabregas, however, ended any hopes of a comeback after 77 minutes when he slotted home after being played in by Arshavin following a surging 40-yard run.