Two goals in a minute in the first half from Diego Godin and Kevin Gameiro allowed Atletico to take control of proceedings at El Sadar before Yannick Ferreira Carrasco struck late on to cap the win.
Atleti had suffered back-to-back LaLiga defeats to Real Sociedad and Madrid before Sunday's game, and they had Jan Oblak to thank for saving an early penalty before Godin opened the scoring nine minutes before half-time.
Simeone highlighted those as the key moments in the victory, but was delighted with their collective display in the second half as they closed out the game.
"It was an important match for us because of the way the game ended up," he told a media conference.
"Osasuna started the first 20 minutes strongly. Oblak was decisive in saving the penalty and with Godin's goal we felt more comfortable.
"In the second half we had a lot of collective strength and organisation and we killed off the match in the end.
"Osasuna never gave in. The result aside, they continued to ask us questions and forced us into making the changes with Saul [Niguez], Carrasco and Thomas [Partey]."
Godin, who headed in a Koke corner for his first goal of the season, was relieved to give his side a lift after a poor run of results that has seen them drop nine points adrift of league leaders Madrid.
"The truth is that I'm happy with the goal and delighted with the win," he said. "We needed a victory in LaLiga and we got it on a very difficult pitch.
"We had nerves that are logical when you're losing. We're used to winning and we lost two games.
"We know what the road is. We had some good feelings, which is what we've needed."