The Swans have picked up only three points on their travels and, although their miserly defence withstood a flurry of chances at the end of the first half, they could not prevent Osman heading in Royston Drenthe's corner on the hour mark.
The visitors had only one chance of note but Danny Graham should have made it count rather than heading straight at Tim Howard, while Swans keeper Michel Vorm twice saved well from Everton substitute Magaye Gueye.
The first half an hour was notable only for a near total lack of goalmouth action, perhaps not too surprising given the scoring records of the two teams.
Everton have found life particularly difficult at Goodison but they took firm control during the final stages of the half, and the catalyst was a forced substitution.
Seamus Coleman, back in the team after a thigh injury, limped off to be replaced by Gueye, with the lively Drenthe switching to the right and setting up the first good chance with a whipped cross that was headed just wide by Louis Saha.
Everton should have taken the lead in the 38th minute when Osman played in Leighton Baines down the left and his pull-back fell perfectly for Gueye.
The Frenchman was too deliberate with his shot, though, allowing Vorm to make a good stop, and Gueye's effort from the rebound was kept out by the arm of Ashley Williams, although it did not look intentional.
Phil Jagielka fired over from the resulting corner and Baines then did likewise with a free-kick after Williams was penalised for handball on the edge of the box.
It was becoming an onslaught, and Drenthe sent a free-kick wide of the far post before the half ended with the influential Osman playing in Gueye only for the winger to again be denied by Vorm.
Swansea boss Brendan Rodgers, who had already lost Neil Taylor to injury, made a change for the start of the second half, sending on Kemy Agustien in place of Mark Gower, and the visitors began to show a bit more attacking intent.
The half-time break had done Everton no favours and they did not threaten Vorm's goal again until the 56th minute, when Phil Neville's quick free-kick found Osman in the box but he could only poke tamely at the keeper.
Four minutes later the Dutchman was beaten, though, with a straightforward goal from a corner.
Swansea may feel aggrieved the set-piece was awarded following a driving run into the box from Drenthe, but there was no doubt about the finish, the winger's delivery planted into the far corner by Osman.
The visitors had barely created a chance but they should have equalised in the 68th minute, Graham making good contact with Wayne Routledge's cross 10 yards out but heading straight at Howard.
Swansea had struggled to cope with the pace of Drenthe and he would have been in again but for a sliding clearance from Vorm.
Rodgers made his final change, throwing on striker Leroy Lita for Leon Britton, while Denis Stracqualursi replaced the disappointing Saha for Everton.
Drenthe received a standing ovation as he made way for Tim Cahill, dropped to the bench after a year without a league goal, with 10 minutes left.
And there were no real alarms for Toffees boss David Moyes as the hosts comfortably held on to leapfrog their opponents and climb to 11th place in the table.