The Al-Qadi family saw their side go behind to Jamie Devitt's penalty when Tom Parkes was penalised for handball.
But Rory Gaffney equalised with a 12-yard volley after away goalkeeper Barry Roche's unconvincing attempts to deal with a Danny Leadbitter cross.
And substitute Billy Bodin's close-range header earned victory late on.
Rovers go sixth, five points adrift of the top three, while Morecambe are in 17th place, 11 points clear of the relegation zone.
Bristol Rovers celebrated their first match under their new Jordanian owners with a victory over Morecambe that takes them up to fifth in League Two.