Fly half Cameron Clement booted four penalties and a conversion for a 14-point haul, with wing Richard Williams and flanker Rob Dudley-Jones scoring crucial second-half tries in a 24-10 scoreline that sees them move to within a single bonus-point win of 11th-placed Neath with five games left to play.
Swansea trailed at half-time after Clement's opening 28th minute penalty was bettered by a converted try from Newport captain Craig Attwell and a Geraint O'Driscoll penalty but they dominated the scoreboard after the interval.
Clement landed three further penalties following the restart and Williams raced away after fielding a poorly-placed clearance kick as Swansea took control.
The impressive Dudley-Jones then finished off proceedings when he crossed nine minutes from time to bring a run of 15 straight Principality Premiership defeats to a timely end.
Swansea have now played a game more than Neath, who weren't in action this weekend, with the gap between the two teams down from nine points to five. Neath travel to St Helen's on the final day of the campaign on April 26, with Saturday's victory meaning that the season finale could now carry even greater significance than just local bragging rights.
The Swalec Cup quarter-finals meant that Swansea versus Newport was the only Principality Premiership tie that went ahead on Saturday but a full fixture list takes place this weekend, starting with Newport's attempt to bounce back at Llanelli on Friday night.