The Swans controlled the game from virtually start to finish against a Scunthorpe side perched just above the relegation zone and went ahead before the half-hour mark courtesy of David Edgar's first goal in over a year.
Shefki Kuqi had been denied by the woodwork before Edgar - on loan from Burnley - broke the deadlock and Ashley Williams added a second for the Welsh side six minutes into the second half.
The bustling Kuqi was rewarded for his efforts 10 minutes from time to put the seal on an impressive Swansea performance.
Victory was Swansea's first at the Liberty Stadium since a 1-0 success over Peterborough in late February and will lift morale significantly heading into the final four games of the regular season.
Swans boss Paulo Sousa responded to Saturday's 2-1 defeat at Cardiff by making six changes to his starting XI, and it was a decision that reaped rich reward as the home side dominated the opening exchanges.
Joe Murphy tipped Angel Rangel's right-footed drive from the edge of the penalty area around the post before Kuqi struck the crossbar after getting on the end of a David Cotterill cross.
Kuqi was then denied minutes later by the Scunthorpe goalkeeper, who also kept out Guillem Bauza's close-range header.
Murphy, though, was powerless to prevent Swansea surging ahead after 27 minutes.
Cotterill's near-post corner was flicked on by Kuqi and Edgar pounced from six yards for his first goal since scoring for Newcastle in their 5-1 defeat against Liverpool in December 2008.
Bauza shrugged off a challenge from Garry Thompson to volley wide from 18 yards and Paul Hayes went close at the opposite end as Scunthorpe searched for an equaliser in the latter stages of the first half.
The hosts continued to boss proceedings after the restart as Williams steered Mark Gower's corner just wide 30 seconds into the second half.
Williams did get his name on the scoresheet after 51 minutes as the Swans doubled their advantage.
Another Cotterill corner caused the visitors all manner of problems and Bauza's wayward shot and some hesitant defending eventually allowed Williams to bundle the ball over the line from close range to take his tally for the season to four goals.
Bauza came close to adding a third some four minutes later, curling an 18-yard effort inches wide, while Cotterill, Joe Allen and Nathan Dyer also created and spurned decent opportunities.
More Murphy heroics kept Kuqi at bay from 12 yards with 22 minutes remaining, but the well-travelled striker did eventually find a way past the Republic of Ireland international after 80 minutes.
Garry Monk's punted pass was flicked goalwards by Niall Canavan and Kuqi seized possession before firing low beyond Murphy.