Sean Devine's 68th minute header settled this relegation dogfight in Exeter's favour but both sides are still in desperate trouble and facing Conference football next season.
Swansea boss Brian Flynn took a positive attitude following his side's devastating defeat.
He said: "I still think one more win is needed, and we have the players at the club who are capable of doing it. I said when we were six points adrift midway through the season that if we came to the last couple of games needing three points to stay up, then we would do it.
"It's a vitally important period for Swansea City but I am sure we will get through it." Flynn was forced to make three changes to the side beaten at Leyton Orient while Exeter included Kwame Ampadu, who spent a number of seasons at the Vetch, and another former Swan Chris Todd was also in the visitors' line-up.
The first half was poor in quality as tension took hold of both sides. Goalmouth incidents were few and far between, although Exeter did have a case for a penalty on the half-hour after the ball appeared to strike Kristian O'Leary's arm from a James Coppinger cross.
After Devine had scored the all-important goal, he had another goalbound effort hacked off the line by O'Leary and then Flack found the net only to have his effort disallowed for a foul.
Exeter comfortably weathered a late flurry from the home side and have now given themselves a chance of staying up.
Exeter's manager, Gary Peters, had nothing but praise for his players after the victory which has lifted them off the bottom of the table.
He said: "We gained a lot of confidence from last week's draw with Oxford and the players are showing a tremendous commitment to the cause. That is typified by Gaia who needed an injection before he could even get out there today.
"It was a gamble for me to use him, but I needed to block the threat from Swansea's right flank and I think it worked."