The hosts, who will hope to compete for a second Spanish top-flight title in four seasons, very nearly got off to the perfect start when Alvaro Morata saw his shot tipped over the bar by a sharp reaction save from Andre Ferreira.
However, just seconds later, the hosts suffered a major blow when captain Koke was forced off with a muscle injury, with Pablo Barrios coming on for the stricken skipper.
After a rather turgid first 40 minutes or so, the game eventually sprung into life late in the half, with Morata heavily involved. Having been booked for handling a ball which he ought to have headed home, the striker was on hand minutes later to net Los Rojiblancos’ first goal of the term.
A ball into the area by debutant Cesar Azpilicueta - who was making his 100th La Liga appearance 13 years after his 99th - was diverted into Morata’s path by former teammate Jesus Vallejo, and the Spain international stroked beyond Ferreira from close range.
There was still time left for the striker to have a second goal ruled out for offside in the final act of the opening period.
Having come close to levelling the score just after the restart, Samu drew Granada level around the hour mark with only their second shot on target when he diverted substitute Gonzalo Villar’s cross past Jan Oblak.
The new boys’ joy, though, was short-lived, as Memphis Depay restored the hosts’ lead six minutes later in stunning fashion with a sweetly-struck effort from range that flew into the top right corner.
Although Marcos Llorente added gloss to the scoreline with a third goal in the eighth minute of stoppage time when he broke free into the area before slotting past Ferreira, Granada boss Paco Lopez won’t be too disheartened by his side’s display in their curtain-raiser.
Diego Simeone, meanwhile, will be satisfied with a victory that sees his team move up to fourth at the earliest of stages, while the result also extends their unbeaten home league record against Granada to 20 matches.
Flashscore Man of the Match: Yannick Carrasco (Atletico Madrid)
