But this win was no fluke and with better finishing in the first half, the Robins could have won by a wider margin.
Tranmere created a couple of early chances with Chris Shuker forcing goalkeeper Shane Higgs to get down and hold a low shot, while Chris Greenacre then latched onto a long ball from Robbie Stockdale but stubbed his shot. However the rest of the first half belonged to Cheltenham.
The Tranmere defence struggled to cope with the pace and power of home striker Kayode Odejayi, who put his team a goal up after eight minutes when he ran onto a pass from Steve Guinan and raced clear before shooting past goalkeeper Gavin Ward.
Odejayi almost made it 2-0 when he robbed Ian Goodison of possession but this time Ward was equal to his shot.
And another forceful run by the Nigerian born forward set up Guinan for a shot that was cleared off the line by Chris McCready.
Cheltenham had yet another good chance to score a second on the stroke of half-time when defender Gavin Caines just failed to make good contact with a driven Grant McCann free-kick at point-blank range.
The second half was more even with Tranmere gaining a greater share of the possession but failing to make it count. Shuker fired wide after Taylor had picked him out with a good ball and Higgs pushed speculative shots over the bar from both Greenacre and John Mullin.
Carl Tremarco saw a goal-bound effort kicked clear by John Finnigan but Cheltenham defended well enough in the second half to keep their lead intact and remain the early season surprise packages.
Their combination of committed defending and pacy attacking looks sure to surprise a few more teams this season.