The forward's finish from a fine Victor Lindelof pass in the first half and a late penalty from Bruno Fernandes secured a commanding first-leg advantage in Thursday's clash at the Nuevo Los Carmenes.
Ole Gunnar Solskjaer's side was not particularly inspired in attack but rarely looked troubled by a team which has already eliminated Napoli in an impressive first European campaign.
Chelsea loanee Kenedy was a threat with shots from distance, but Diego Martinez's men were unable to create clear openings against a diligent Red Devils rearguard, particularly with only 33 per cent of the ball.
Rashford, who sustained an ankle problem in the win over Brighton and Hove Albion last Sunday, appeared to be struggling in the early exchanges as United made hard work of breaking down Granada.
The England forward looked fit and sharp 31 minutes in, though, latching onto Lindelof's long ball over the top before prodding a composed finish beyond goalkeeper Rui Silva.
The goal sparked a little life into Granada's attack, Kenedy's ambitious volley from the edge of the area giving the returning David de Gea something to do.
Kenedy twice tested De Gea again from outside the box after the restart but they were comfortable saves for the United keeper, who was otherwise mostly untroubled.
Fernandes had had a quiet game but he squeezed a mishit penalty under Silva's grasp to score a potentially decisive second away goal in the 90th minute after Yan Eteki was penalised for pushing a hand into the Portugal star's face.