A second-half strike from Andy Roscoe was enough to give Exeter a vital victory against their fellow strugglers.
Exeter had the best early moment when Barry McConnell met James Coppinger's cross but he headed weakly into the arms of the goalkeeper.
The Grecians came close to taking the lead in the 17th minute.
Roscoe found Martin Thomas and the former Swansea midfielder skipped past a defender before striking a fine shot against the post from the edge of the box.
Coppinger went close midway through the half when Steve Flack knocked down Roscoe's cross and his low shot from the edge of the box just fell the wrong side of the post.
Matt Murphy shot into the arms of Kevin Miller from 15 yards as Swansea pushed forward and Leon Britton, making his debut on-loan from West Ham, could also only find the goalkeeper with a shot from the edge of the box.
Thomas should have put Exeter ahead on the stroke of half time when he met Roscoe's cross but he headed the ball over the bar from close range.
After the break both teams chased the game in search of what could prove a crucial victory.
The game was won in bizarre circumstances. Santos Gaia was judged to have been obstructed inside the box and the referee signalled an indirect free-kick.
Carl Pettefer tapped the ball to Roscoe who hammered it into the bottom corner.
James Thomas almost responded instantly for Swansea when he struck a long-range shot just over the bar.
In stoppage time Thomas came close to stretching the lead when he skipped into the box but saw his powerful shot charged down.
It was not pretty stuff but it was the perfect result for the Grecians.
Exeter boss Neil McNab said: "This was a massive win for us.
"We showed a lot of courage in the second half and it was a really gutsy performance."