The Toffees toiled for much of the match before a second-half header from a Royston Drenthe corner gave the hosts a deserved victory over Brendan Rodgers' determined outfit
The visitors had fought well and looked on course to grab their fourth away draw of the campaign before switching off needlessly at a corner kick.
Royston Drenthe duly delivered perfectly for Osman to leap highest above the Swans defence 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 whilst 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 their 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 their lines.
The hosts 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 they served to underline why - with one exchange resulting in a cross from Gower which was gratefully collected by Howard.
Any attempts to break the Everton line were subsequently repelled and one attack even saw Everton’s Drenthe – who has been recently criticised 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 Fellaini at the other end but saw his cross cut out desperately by a sliding Leon Britton.
The former Real Madrid man was proving the only truly inspirational figure in the Toffees’ attack but his task was underlined after a run, which left two Swansea defenders standing, saw him meet a wall of Swansea resistance with little support in Blue.
Coleman’s opportunity to impress then came to a premature end as he limped off on the half hour to be replaced by young prospect Gueye.
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.
Gueye had a golden opportunity to put his side ahead before the break after a clever ball by Saha but failed to fire home after Vorm had made himself big and ensured the sides went in level.
The visitors made a change at half time with midfielder Gower, who had been key to their solidity in the opening 45 minutes, making way for Kemy Agustien.
Everton failed to get fully back in their stride for the opening periods of the second half and were restricted to a surge from defence by Jagielka which resulted in a poor left-footed cross.
Their 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 Vorm after a heavy first touch.
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 leapt above the Swansea defence before nodding home adroitly.
Jagielka soon went close with his own header from a corner after he stooped in front of the Swans defence but failed to put his effort beyond the front post.
Swansea needed a chance to emphasise the vulnerability of a one-goal lead and they found it when Wayne Routledge crossed for Danny Graham only to see the striker nod it straight at Tim Howard.
The forward looked disappointed with his effort and had every right to be as replays showed that had it been a yard either side of the goalkeeper it would have resulted in a goal.
The hosts pushed for a second goal to give themselves some comfort with David 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.
Though the substitutions didn't bring a second goal, they allowed the hosts to starve their opponents and substitute forward Leroy Lita of possession and allowed them to hold on for a deserved victory.