A month had passed since Atleti's previous top-flight triumph over Espanyol on 11 November (AEDT) and the brilliance of Osasuna goalkeeper Sergio Herrera looked like spelling another frustrating outing at Wanda Metropolitano.
But Morata dispatched an emphatic header midway through the second half to break the deadlock and Saul Niguez doubled the advantage 15 minutes from time.
Los Rojiblancos move up to fourth in the table, six shy of Barcelona after the leader was held to a 2-2 draw at Real Sociedad earlier in the day.
Herrera began a busy first half by thwarting Morata at close quarters before the former Juventus striker turned provider and played in Joao Felix in the 23rd minute, the Osasuna keeper saving with his boot.
A double stop from Herrera spared Jagoba Arrasate's side 10 minutes from half-time, parrying a shot on the angle from Thomas Lemar and getting back into position to repel Koke's follow-up drive.
The pick of Herrera's work came in the 39th minute when, at full stretch, he turned Joao Felix's header from teenage debutant Manuel Sanchez's left-wing centre onto the crossbar.
He was not done there, keeping out Thomas Partey's rasping long-ranger and frustrating Joao Felix once more to ensure parity at the interval.
Morata got behind the visiting defence only to miss his kick woefully and Herrera's masterclass continued as he saved with his foot from Sanchez.
Avila's sweetly struck volley kept Jan Oblak honest in the Atleti goal before Sanchez was finally beaten in the 67th minute.
Trippier delivered a typically threatening set-piece from the right and Morata put his earlier profligacy behind him to power home.
Saul sealed the points in style, darting forward from the unfamiliar position of left-back to dink over Sanchez, Simeone letting out a guttural roar on the touchline.