Leading the line in place of Antoine Griezmann, Sorloth opened the scoring in the 23rd minute. He latched on to a long ball from goalkeeper Jan Oblak and dispatched a half-volley that squirmed past Simon Mignolet.
Brugge responded nine minutes before half-time. From a corner, Brandon Mechele flicked on at the near post and Joel Ordonez headed home.
Ordonez, who had scored an own goal in the chaotic 3-3 first leg when Atletico surrendered a two-goal advantage late on, made amends to level the tie once more.
The visitors almost went in ahead, but Hugo Vetlesen was denied by Oblak after meeting a cross from Christos Tzolis.
Three minutes after the restart, United States international Johnny Cardoso rifled in a low strike for his first Atletico goal, putting the hosts 5-4 ahead on aggregate.
Sorloth then took centre stage in the closing stages, scoring twice in the final 14 minutes to seal victory and end Atletico's three-game winless run in the competition.
Having also netted twice against Espanyol in La Liga last weekend, the Norwegian's surge in form sets up a last-16 tie against either Tottenham Hotspur or Liverpool.
"I'm very happy, there's a lot of hard work behind these results, which are starting to be more positive than negative," Simeone said.
"Many new players joined us last season and many more this season, so reinventing ourselves is never easy, and when you see the willingness and conviction of the substitutes when they come on, that moves me.
"I'm a football man, and when I see that commitment to the idea, to what we're looking for, I feel very identified with it."
FK Bodo/Glimt produced one of the biggest upsets in Champions League history by eliminating Inter Milan at the San Siro to reach the last 16 for the first time in their debut campaign.
Diego Simeone has urged Atletico Madrid's supporters to be the driving force behind his side as they prepared for their UEFA Champions League knockout round play-off second leg against Club Brugge KV at the Metropolitano Stadium tonight.
Benfica winger Gianluca Prestianni has been provisionally suspended for one match by UEFA following allegations of racist abuse directed at Real Madrid forward Vinicius Junior during last week’s Champions League first-leg tie in Lisbon.
Vinicius Junior got on the scoresheet for Real Madrid but it was not enough to prevent them falling to a shock 2-1 defeat to Osasuna at Estadio El Sadar in LaLiga on Saturday.
