The Toffees toiled for much of the match before a second half header from a corner gave them a deserved victory over Brendan Rodgers' determined outfit.
Royston Drenthe delivered perfectly for Osman to leap highest above the Swans defense and grab what would prove to be the winner.
Everton boss David Moyes made two changes, notably in attack, from the home draw with newly-promoted Norwich.
Drenthe, so impressive in his substitute appearance in that game, replaced Magaye Gueye on the left wing while Seamus Coleman returned at the expense of Tim Cahill.
Meanwhile, Brendan Rodgers left his side unchanged from a hard fought draw with Newcastle United in its last outing expecting an equally determined performance.
Everton’s attacking changes seemed to be working early on as Drenthe’s pressing game and direct running were evidenced and he was unfortunate to see a cross go straight out of play after some fine work up against two defenders.
Moments later, Coleman found space on the right but failed to find a fitting cross after a typically clever run, allowing Swansea to clear its lines.
The home side kept up the pressure as Leighton Baines’ free kick barely evaded the boot of Louis Saha after the striker pulled off his man in the box.
Swansea’s passing football has gained plaudits throughout the Premier League this season and the club underlined why with one exchange resulting in a cross which was gratefully collected by Tim Howard.
Any attempts to break the Everton line were subsequently repelled and one attack even saw Everton’s Drenthe – who has been recently criticized for his work rate by boss Moyes - back on his own byline to make a neat sliding tackle.
The Dutchman was then released brilliantly by Marouane Fellaini at the other end but saw his cross cut out desperately by a sliding Leon Britton.
More frustration followed as Saha nodded wide from a Tony Hibbert cross before Baines found space in behind to pick out Gueye whose attempt to side-foot home brought a fantastic reaction stop from Michel Vorm.
Swans left back Neil Taylor was withdrawn for Angel Rangel after picking up an injury in the ensuing goalmouth scramble and Phil Jagielka wildly hacked at the resultant corner to send it over the bar.
Everton failed to get fully back in its stride for the opening periods of the second half and was restricted to a surge from defense by Jagielka which resulted in a poor left-footed cross.
The next chance fell to Osman after some sloppy Swansea defending allowed the midfielder into the box unmarked but he could do little more than toe-poke weakly at goal.
However, the Englishman quickly atoned for his poor effort as he put his side in the lead. A Drenthe corner allowed Osman to get free and he jumped above the Swansea defense before nodding home adroitly.
Swansea needed a chance to emphasise the vulnerability of a one goal lead and the club found it when Wayne Routledge crossed for Danny Graham only to see the striker nod it straight at Howard.
The forward looked disappointed with his effort and had every right to be so after replays showed that had it been a yard either side it would have resulted in a goal.
The home side pushed for a second goal to give itself some comfort with Moyes looking to the bench bringing on Denis Stracqualursi for the tiring Saha.
Drenthe was then withdrawn to a deserved rapturous ovation as Tim Cahill provided fresh legs in the Blues’ attack.
The moves would help Everton see out the match and earn three vital points.