Stand-in captain Osman's thumping drive set Everton on their way to three points before Nikica Jelavic scored an injury-time second after Steven Pienaar had been sent off.
Following last week's shock FA Cup quarter-final defeat at home by Wigan Athletic, Everton are likely to need a fifth-placed finish to qualify for Europe and Osman is staying positive.
"It's difficult to play with 10 men against any team, let alone the champions, but we kept them at bay and I don't think they had too many chances to score, which shows how dogged we were in defence," Osman told Sky Sports.
"The league's what we've got to concentrate on now and get as high up as we can. That's an aim that's important to us and we want to win all our remaining games."
Osman was delighted with his 30-yard strike which swerved away from City goalkeeper Joe Hart, saying: "Joe's a fantastic keeper and to beat him from that range, I can't ask for more, I'm delighted.
"It maybe bobbled a bit as I hit it, but it went in the back of the net and gave us the goal that we needed to try to go on to win the match."
Seamus Coleman was an impressive figure at right-back for Everton and he hopes the victory over City helps to make up for the FA Cup exit against Wigan.
"It was disappointing to go out of the FA Cup and the manager got us on it this week," said Coleman.
"We tried to repay the fans as much as we could and I think we did that. Winning games is what it's all about."