Midfielder Tom Elmes — a former minor and Under-21 footballer with Wexford — fired the all-important goal in the 55th minute.
Pat Dolan’s side have now grabbed four points from their opening two games, while Ramblers remain pointless at the bottom of the First Division with Kilkenny City.
Meanwhile on Saturday, Derry City joined St Patrick’s Athletic at the summit of the Premier Division with a 2-0 win over Sligo Rovers at The Showgrounds.
Barry Molloy fired the Candystripes ahead and Kevin McHugh doubled the advantage in the first half.
It could have been worse for Sligo, too. After nine minutes Keith Foy and Jason McCartney were involved in a mix up and McCartney felled the onrushing Gareth McGlynn inside the box. Referee Dave McKeon immediately pointed to the spot.
Mark Farren took the resultant penalty but Rovers goalkeeper Richard Brush dived left to save.
But the goals inevitably came. Molloy fired the first on 34 minutes, which deflected past Brush.
Five minutes before half time McHugh persistence paid off as he grabbed his third goal of the season.
His first effort was blocked but he reacted quickest, lobbing the ball over keeper Brush to put Derry two ahead.
Sligo came back strongly in the second half, with defender Keith Foy’s free kick testing keeper Pat Jennings.
Derry held out but there was a sour note for Pat Fenlon’s side late on, as Dave Rogers saw red for his desperate lunge at Chris Turner after 86 minutes.
Derry are facing into an important week. Tomorrow they face Linfield in the Setanta Cup at Windsor Park, before they face St Pat’s on Friday night at the Brandywell in an early top-of-the-table clash.
Elsewhere, Monaghan United snatched a 1-1 draw with Kildare County at Station Road in the First Division on Saturday night.
The Thoroughbreds led through Philly Gorman’s 14th-minute goal only for Monaghan to grab an equaliser on 83 minutes, as Robbie Farrell converted from the spot after Brian Gannon was taken down.
Monaghan held on to leave Kildare frustrated by their poor finishing. Both sides stand mid-table with two draws from their opening fixtures.