The Saints gave the defending English champions a scare in the second half in their opening game of the new Premier League season when they went 2-1 up.
Goals from Edin Dzeko and Samir Nasri eventually saw City win maximum points at home, but Adkins was happy with his players' display.
"We've come away from home to take on the champions and we tried to find a way we could win this game," he said.
"That's what we tried to do and our defence have just faced the best forward line they've ever come up against and they didn't do too badly.
"City conceded only 12 goals at home last season and we've just scored twice. They only dropped two points at home and we pushed them hard today, so our lads can take a lot from this game."
Southampton are back in Premier League action next Saturday when they host Wigan Athletic.