The hosts had the better of the attacking play but Neil Wood’s side, and Smith in particular, were clinical with the few clear-cut opportunities they had to secure a first victory in four matches.
It took just two minutes for Salford to open the scoring when Smith sent a header into the ground and in off the crossbar.
Doncaster – looking for a fifth consecutive home win – pushed for an equaliser and Harrison Biggins’ audacious chip was kept out by the fingertips of Alex Cairns.
Smith doubled the visitors’ lead in the 26th minute with a trademark header as he powered home from an Ethan Ingram cross.
Joe Ironside looked to have pulled a goal back for Rovers in the second half but his header was ruled out for offside, while Zain Westbrooke and Luke Molyneux both hit the bar.
Smith completed his hat-trick in the 77th minute as he pounced to slot in after substitute Kelly M’Nai was denied by Louis Jones.