After a dreadful first half, City took control in the second half, with Mario Balotelli scoring his first league goal of the season and James Milner chiming in with a screamer.
"We found it quite tough going in the first half, we had some stray passes but we ground it out in the second half and that's a great three points," Barry told Sky Sports.
"Credit to Wigan, we had to change things to get past them."
The goals came after some City fans had jeered boss Roberto Mancini's decision to substitute striker Sergio Aguero for defender Aleksandar Kolarov.
But Barry says the manager was proved correct and that City will have to take the rough with the smooth: "When the game is going our way the manager is the best tactician there is, that's the world we live in.
"A lot of teams come to Wigan and don't get results. These are vital points that have come our way."