The Toffees suffered a shock 3-0 defeat at home to Roberto Martinez's men to end their hopes of silverware this season and were booed off at full-time.
But they bounced back tremendously at Goodison Park today, with Osman setting them on the path to three points with a cracking opener in the 32nd minute.
Steven Pienaar was dismissed just after the hour, which meant Everton were on the back foot for most of the second half, but Nikica Jelavic finished off a counter-attack in injury-time to seal victory for Everton.
Saturday's win was just the tonic for Coleman, who told Sky Sports 2: "It's what we aimed to to do. Ultimately disappointing to go out of the FA Cup. We went out there and tried to repay the fans as much we could and I think we did that.
"Winning games is what it's all about and it's never nice to lose games, especially when we thought we could do well in the FA Cup this year.
"But we still have a lot to play for in the league and we showed it there today [Saturday].
"That wasn't the Everton team last week, we just didn't turn up.
"Going down to 10 men we just worked as hard as we could. We're all mates here and I think that showed."
Pienaar was sent off for two bookable offences and Osman was thrilled with the way they managed to deny City, whose slim hopes of regaining their Barclays Premier League crown are in tatters.
"It's difficult to play with 10 men against any team, especially the champions," he said.
"We kept them at bay and I they didn't have many chances to score, which just goes to show how good we were in defence."
City could have had a penalty late on when Marouane Fellaini was penalised for handling a Carlos Tevez shot, although Osman preferred to focus on using this result as a platform to finish strongly in the league.
He said: "It's what we've got to concentrate on now, the league and get as high up as we can. And I think we showed today it's important to us and we want to win all our remaining games."