As early as the second minute Molyneux made sweet contact with a 30-yard free kick and Rovers keeper Steve Mildenhall could only palm the shot into the roof of the net.
Rovers were unable to summon up a meaningful response and it was Stanley's 131 supporters in a crowd over over 7,000 who were celebrating along with their team at the final whistle.
Molyneux's goal was one of few moments of excitement in a scrappy first half, which saw Accrington defend strongly.
It took Rovers 23 minutes to register a shot and even then Tom Parkes' effort was straight at goalkeeper Paul Rachubka.
Young midfielder Ollie Clarke, making his first start for the home side, put in their best first half effort just before the break when his dipping volley narrowly cleared the crossbar.
The second half was a similar story with Rommy Boco heading wide in the 53rd minutes, passing up a great chance to double Accrington's lead.