Ian Morris' goal shortly after the hour mark wrapped up the points as the visitors made it seven straight wins to move up to third in League Two, where a solitary point now separates the top five.
The Irish midfielder stabbed the ball over the line after Kevin Nicholson's corner had been headed into the danger zone by Brian Saah.
Cheltenham finished the half strongly with Kaid Mohamed having a long-range effort saved by Gulls 'keeper Robert Olejnik, who then did well to punch clear a fierce Marlon Pack free-kick on the stroke of half-time.
The Robins had a glorious chance to level 80 seconds after the restart when a jinking run from Mohamed took him into the area only for Olejnik to deny him with a brave save.
The visiting stopper then pulled off a brilliant reflex save 20 minutes from time to deny Russ Penn, the giant Austrian getting down to palm away the shot from 12 yards. Substitute Jeff Goulding was denied by Saah's last-gasp clearance in injury time as the Gulls held on for victory.
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