The Gunners had shown plenty of spirit and glimpses of the quality they will need to have an impact on this season's title race during an entertaining encounter with the Italian champions.
Then just when the match seemed to be heading towards a draw, van Persie - out since breaking his metatarsal in January - smashed home a winner following superb close control in the six-yard box.
As expected, manager Arsene Wenger made several changes from the starting XI for Saturday's victory over Paris St Germain.
Those changes included bringing in centre-back William Gallas, who was named captain ahead of Kolo Toure, and handing a first-team debut for teenage winger Kieran Gibbs.
Summer signing Eduardo da Silva was still missing as he awaits next week's appeal to the Home Office for a work permit.
Inter were very disappointing against Valencia on Saturday, and having also rotated several of their players, the Italians offered little threat during the opening exchanges.
Van Persie looked lively in attack for the Gunners and was ably supported by Alexander Hleb.
On 20 minutes, a clever step-over from Hleb created a shooting chance just outside the box for van Persie but Julio Cesar forced away his low effort.
Former Everton defender Marco Materazzi was cautioned for a late challenge on Emmanuel Eboue as the Arsenal defender, playing on the wing, surged down the right flank.
From the resulting free-kick, a yard outside the area, van Persie forced a good reaction save from Cesar at his near post.
Mathieu Flamini - on target against PSG on Sunday - made the Inter keeper work again to stop his low 20-yard strike.
Summer signing Bacary Sagna also saw his angled shot saved as Arsenal and van Persie shot over from a free-kick.
Wenger kept the same XI for the second half, with Eboue again dangerous down the right.
Arsenal felt they should have been awarded a penalty when van Persie went down following what looked like a push by defender Cristian Chivu, a big-money summer signing from Roma but referee Mark Halsey was unimpressed.
Van Persie needed brief treatment on his shoulder following the awkward fall but was soon up and running again.
Out of nothing, Inter snatched the lead on 52 minutes.
Cesar the midfielder floated the ball over from the right and Honduras striker David Suazo got above Gallas to power a header in off Jens Lehmann's right-hand post.
Eboue smashed an angled drive over as Arsenal looked to quickly get back on level terms.
The equaliser came on 68 minutes through Hleb. Following a break down the right, Sagna cut the ball back to the penalty spot.
Francesco Toldo, on at the break for goalkeeper Cesar, made a brilliant save to keep out Hleb's first effort, but had no chance of stopping the low shot into the corner from the rebound.
The Inter keeper produced another fine reaction stop to deflect away van Persie's close-range effort flowing a low right-wing cross from Hleb.
Young Danish striker Nicklas Bendtner came on for Gibbs to add some weight to the Arsenal attack in the final 10 minutes.
Yet it was van Persie who the game late on. There looked little danger when Eboue fed the Dutchman, surrounded by blue shirts, on the right side of the area.
However, van Persie produced a sublime dragback to turn defender Daniele Pedrelliez inside out - before smashing the ball high into the roof of the net.