Needing a win to put itself level on points with table-topper Real Madrid, Quique Setien's side started sluggishly and would have gone behind if not for VAR (video assistant referee).
Barca made its good fortune count, though, Lionel Messi — on his 350th official appearance at Camp Nou — setting up Griezmann to open the scoring in the 33rd minute.
Sergi Roberto doubled Barca's tally before half-time, but Barca-linked Angel Rodriguez set up a grandstand finish, with Marc-Andre ter Stegen just managing to deny Getafe an equaliser as the champion held on.
Jordi Alba succumbed to a right adductor injury in the 22nd minute, with Junior Firpo introduced, and things looked to have got worse for the hosts when Allan Nyom tucked home from a corner, only for the goal to be ruled out by VAR for a foul on Samuel Umtiti.
Marc Cucurella was inches away from putting his parent club behind moments later, yet it was Barca who broke the deadlock when Messi fed a wonderful, disguised pass through to Griezmann, who finished with a sublime flick over David Soria.
It was 2-0 six minutes later, Roberto sending a first-time finish into the bottom-right corner at the culmination of a slick move, and only a fine save from Soria prevented Messi adding a third before half-time.
Angel was brought on shortly after the restart and the change had the desired impact when he volleyed in from the edge of the box.
Ter Stegen denied Jaime Mata a tap-in with 18 minutes remaining and, although Griezmann squandered two great chances, Barca ultimately made its goalkeeper's save count as it claimed the points.