Neither side enjoyed a particularly bright start to the game but Hume gave Rovers the lead after 24 minutes. The Canada international, on loan from Preston, latched on to Martin Woods' cross from the left and took advantage of hesitation between Pompey goalkeeper Mikkel Andersen and skipper Brian Howard to slide the ball home.
The hosts showed more energy in the second half and Lee Williamson should have equalised after 67 minutes but he miscued his effort following a cross from all-action loanee Nathaniel Mendez-Laing.
Doncaster came close to doubling their advantage two minutes later when Rob Jones' header from David Cotterill's corner drew a superb stop from Andersen.
Izale McLeod wasted a glorious chance to level for Pompey after 79 minutes as he raced on to Howard's throughball but he lobbed his finish over the crossbar.
A Gabor Gyepes header then whistled just over in the third minute of stoppage time as the Yorkshire club ended a run of two consecutive losses in League One.