The Englishmans 6-footer for birdie completed a 3-under 69 for a 54-hole total of 200.
Graeme McDowell shot a 71 for 203, including a bogey at the last hole.
South African David Frost also shot 69 on a windy afternoon to hold third place at 206, with Michael Jonzon (71), Paul McGinley (71) and Soren Hansen (72) sharing fourth at 208.
Fishers drive at the 471-yard 18th just cleared the lake on the left that McDowell hit into. Fishers 9-iron finished 6 feet from the pin and he made the only birdie of the day on that hole by sinking the putt.
McDowell took a penalty and hit his third into a greenside bunker on the way to a 5.
Its always nice to finish the day on a good note and it is rewarding to shoot 3-under when I was 2-over after seven holes, Fisher said.
I told myself to stay with it, stay patient, he added.
Fisher birdied the next two and eagled the downwind par-5 15th with a huge drive and a pitching wedge to 145 feet.
McDowell also eagled 15 and was tied with Fisher at that point, but he bogeyed the 16th and 18th.
Im obviously disappointed with my finish but Ross and I had a lot of fun out there, McDowell said. I had a simple up and down on 16 that I made a mess of. Things can turn around out there very quickly when the wind is blowing like that.
Frost, who is 48, credited his 69 to experience.
You just want to try to keep hitting fairways and the right side of the greens, said Frost, whose only bogey came at the 18th, when he found the deep pot bunker in front of the green.
If you had told me Id only have one bogey in this wind, Id have taken it, Frost said.
McGinley was content with his 71 in the conditions.
A 71 is good in this wind. Ive had my foot to the ground for three days. Ive had three good scores, McGinley said. A 71 is a good score this afternoon but Ive made no ground on the leaders'in fact, Ive lost ground.
McGinley is among some six players with a chance of taking the one spot available in the British Open in 11 days with a top five finish. Others with the same chance are Frost, Jonzon, Jeev Milka Singh, Stephen Gallacher and Rory McIlroy, who shot 72 and shares 10th place.
Sergio Garcia slipped to a 74 and dropped from third to a tie for seventh.
Colin Montgomerie bogeyed the last three holes for a 73 and is tied for 10th.