The Royals, who had lost their last two matches to drop down to fifth, returned to winning ways courtesy of a Roy Beerens goal early in the second half of a match that was originally abandoned at half time at the end of December due to heavy fog.
It was Reading who were the first to threaten the opener when Garath McCleary beat the offside trap to race down the left flank, but the forward's subsequent low cross into the box was too close to Fulham keeper David Button.
Clear chances were proving tough to come by for both sides, but it was Button who was the busier of the two keepers in the first half and he was called upon shortly before the half-hour mark to deny Swift after he had been played in by Beerens.
Fulham's best opportunity in the opening 45 minutes came with just over 10 minutes remaining until the interval when Sone Aluko found himself in a good position, but under pressure from Liam Moore he fired his resulting effort over the crossbar.
Reading upped the pressure in the final five minutes of the half as Swift nodded over after good work from Beerens, who then came close himself with a low drive that needed to be turned around the post by Button.
Fulham were able to hold on until half time, but they didn't last much longer upon the restart, with Reading breaking the deadlock just three minutes into the second half. Moore was adjudged to have been brought down inside the area and, while Swift saw his resulting penalty saved, Beerens was following in and tucked home the rebound to save his teammate's blushes.
The hosts continued to pile the pressure on when searching for a quickfire second, with Paul McShane and Swift coming close before McCleary rattled the crossbar and saw the ball bounce down in front of the line.
McCleary was then denied by the keeper after good work from Moore before the busy Button made a more routine stop to keep out a curling effort from Danny Williams.
It looked as though Reading would pay for their missed opportunities when Fulham were awarded a penalty in the final minute of normal time, but Chris Martin - scorer of four goals in his last four league games against Reading and five in his last six Championship outings for Fulham - failed to continue that streak when his powerful spot kick was kept out by Ali Al-Habsi.
There was still more work for Al-Habsi to do as he collected a looping volley from Scott Parker in stoppage time, but Reading were able to hang on for a vital three points that lifts them back up to third in the table, now nine points adrift of the automatic promotion places.
Fulham, meanwhile, miss the chance to move within two points of the playoff places and remain ninth in the table.