The hosts scored the only goal of the game after 19 minutes when Myles Weston was felled by keeper Conrad Logan as he raced towards goal.
Alexander made no mistake from the spot but it was Bees keeper Richard Lee who kept his side in the game with a string of second-half saves to deny Rovers livewire striker Will Hoskins.
It was all Brentford, though, before the break as Weston teed up Alexander at the far post only for his goalbound header to be chipped wide at full stretch by Logan.
Hoskins could have sent the Pirates in level at the break but his close-range volley flashed inches over.
After the interval it was almost all Rovers as they piled forward in search of an equaliser, the best chance falling to Hoskins who glanced his close-range header agonisingly wide. He was denied twice more in one-on-ones with Lee, while at the death Brentford's Nathan Byrne and Alexander could have put a more emphatic gloss on the scoreline but were denied by Logan.
Source: PA
Source: PA