Iffy Onuora's team have now gone six games unbeaten as they looked to haul themselves off the foot of the League One table.
The best chance of the game fell to Swans substitute Marc Goodfellow who picked up a cross from Roberto Martinez and with a quick turn, fired a drive which crashed against the crossbar.
Town keeper Rhys Evans was outstanding throughout denying Goodfellow and Adrian Forbes to keep the score at a stalemate.
Swansea will leave the County Ground slightly aggrieved that their possession and general attacking play was not rewarded with more than the one point they picked up.
In a dull first half, Swindon nearly took the lead on the stroke of half-time when the impressive Ricky Shakes crossed into the penalty area for Aaron Brown, but the on-loan QPR midfielder saw his goalbound effort well blocked by Sam Ricketts.
Shakes again caused problems five minutes into the second half when his inviting cross into the six-yard box failed to find any Swindon attackers.
The away side, backed by 3,500 travelling fans, saw a frustrating afternoon compounded when star striker Lee Trundle was forced off on the hour mark with an ankle ligament injury which could keep him out for up to a month.
The game, however, seemed to come to life in the last 20 minutes. First Trundle's replacement, Adrian Forbes, saw his low shot well saved by Evans after good work by Adebayo Akinfenwa.
Andy Robinson then was unlucky when his chip three minutes from time was well saved again by Evans, who tipped it over for a corner.
Swindon substitute Charlie Comyn-Platt nearly won it for the Robins when Shakes outstripped the Swansea backline to cross, but somehow the former Bolton youngster put the ball wide from just two yards out.