The 29-year-old Argentine beat the old mark of 177 set by Eric Brook during the 1930s when he netted City's third goal in Naples after 69 minutes.
And West Brom midfielder Barry, who became the Premier League's record appearance-maker after playing in his 633rd game in September's defeat at Arsenal, reckons his former team-mate will be considered one of football's great forwards when he hangs up his boots.
With Napoli having pegged City back to level at 2-2 from the penalty spot on 62 minutes, Aguero restored his side's lead seven minutes later with a cool finish from just inside the penalty area following a lightning counter-attack.
Frello Filho equalised before Sergio Aguero and Raheem Sterling secured all three points. Insigne sped down the inside left flank, passing to Mertens who, with his back to goal, flicked the ball delightfully back to his onrushing teammate who then calmly curled the ball into the back of the next.More news: Samsung Unveils Business-First Galaxy Note8 Enterprise Edition
The visitors had a super chance to level the scores in the 32nd minute when Aguero found space inside the Napoli box after a fast break, but the Argentine's effort hit Raul Albiol before deflecting just wide of Pepe Reina's post.
Napoli cleared the resulting corner behind but conceded from City's second attempt as Stones powered in Sane's outswinger via the underside of the crossbar.
Ederson saved a spot-kick from Mertens in Manchester but Jorginho stepped up to calmly send him the wrong way.
It was also a goal to finally crumble Napoli's resolve and they looked a beaten side before Sterling made sure on the end of a typically impeccable De Bruyne pass.