The Frenchman's 60th-minute thunderbolt, before he laid on a second for the Argentine in the 86th minute, ensured the English champions did not suffer another frustrating early exit in Group E.
City's great escape into the last 16 seemed unthinkable when they trailed in their penultimate game at home to Bayern Munich with just five minutes remaining before Sergio Aguero completed his famous hat-trick.
Victory or a score draw offered City's most realistic chance of progressing from the group behind Bayern Munich in a fevered atmosphere in Rome and even without their injured totem Aguero, Manuel Pellegrini's men produced a performance full of resolve.
Nasri, given too much space on the edge of the box, hammered a glorious 20-metre shot which cannoned off the post and into the net, and City defended manfully before Nasri freed the rampaging Zabaleta to score his first Champions League goal.
"People wrote us off without our key players (like Aguero and Vincent Kompany) but we're a strong unit, we believed in what we could do," City keeper Joe Hart told Sky Sports.
"You don't win two Premier League titles in three years easily. We have resolve, character and fantastic players. Add that together and you've got a chance."
With various permutations at the start of the night leaving it possible for Roma, City or CSKA Moscow to join Bayern in the last 16, the game effectively became an Anglo-Italian shootout once news of Bayern's early lead over the Russians filtered through.
The English champions had their moments in the first half, especially when James Milner twice forced sharp saves from Morgan De Sanctis.
Hart then stepped up with some crucial stops from the moment he spread himself to block Jose Holebas's fifth minute shot from point-blank range.
As the hour mark approached, Roma seemed to be running out of ideas when the in-form Nasri celebrated his 50th appearance in the competition with a marvellous goal.
Roma, by now needing to score twice, came closest when Hart made another crucial intervention to push Kostas Manolas's header on to the post and Martin Demichelis blocked a shot on the line before Zabaleta calmed City nerves.
Roma had the minor consolation of qualifying for the Europa League, ahead of CSKA Moscow who lost 3-0 at Bayern in their final group match.
(Reporting by Ian Chadband, editing by Ed Osmond)
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