Depor supporter Jimmy Romero was viciously beaten and left in the Manzanares river in Madrid by Atletico ultras 11 years ago, dying as a result of his injuries, and he was remembered by a banner behind one goal at the Galician side's Riazor stadium.
The cup match had been declared a high-risk game, and Spanish media reported there were some moments of tension between supporters outside which did not develop into serious incidents.
Off the pace in La Liga and after defeat by rivals Real Madrid in the Spanish Super Cup semi-finals last week, a win was vital for Atletico to boost their chances of some silverware this season. Simeone's side ended up with nothing in their frustrating previous campaign.
French forward Griezmann struck the woodwork towards the end of the first half as Atletico dominated but could not find a way through.
Diego Simeone hooked out-of-form striker Julian Alvarez before the hour mark, with the Argentina international having scored only once in his last nine matches.
Shortly afterwards, Atletico took the lead, with Griezmann delicately curling a free-kick into the top right corner from just outside the area, and it proved decisive.