The Portugal striker missed an open goal in the early stages of the game before giving his side the lead at the end of the first half from the penalty spot, but then squandered a spot-kick at the start of the second period, hitting the post.
The €126 million player soon made amends by getting Atleti's second, blasting into the top corner to complete a sensational counter-attack and score for the fourth time in two games after his double in Tuesday's 3-2 Champions League win over RB Salzburg.
"We have to keep going," Felix said after the match. "We're doing really well. Personally I'm happy with the two goals and collectively too because we played a great game. The first half was even, in the second we started well and we got the win.
"I've started the season well, I'm working hard, my teammates are helping me and I want to carry on like that."
Osasuna did manage to strike back with a header from substitute Ante Budimir with 10 minutes to go, but Atleti soaked up the pressure well and clinched the points with a late strike from Lucas Torreira, who superbly controlled a cross from Kieran Trippier on his thigh before smashing into the roof of the net.
After the match, Diego Simeone said of Felix: "He's always had important moments, what he's showing now is consistency in his play and hard work. Last year we talked about transition and you were surprised. Little by little you're seeing that work."