Atletico Madrid led 2-0 after the first leg of their last-16 tie but Cristiano Ronaldo's hat-trick on Tuesday sent Juve into the quarter-finals.
Reports had suggested Juve were considering firing coach Massimiliano Allegri ahead of Tuesday's second leg after a row with the club's board.
But Allegri answered his critics as Ronaldo and Juventus producing a superb performance that saw Atleti fail to record a single shot on target in Turin.
"This is a great night for the Juventus fans," Bonucci said to Sky Sport Italia.
"Many things were said, both after that game [the first leg] and over the last few days, but Juve always respond on the pitch.
"We had four chances and put three of them in. They did the same in the first leg, having two chances and scoring both.
"We put our cojones into this match. Atleti were missing players, but so were we, and Leonardo Spinazzola put in a great performance in an incredible atmosphere."
Bonucci may have been referring to Atletico boss Diego Simeone's first-leg celebration which saw him fined €20,000 last week, with the Argentine having previously explained the gesture was directed at Atletico fans to show them that the team "have cojones".
Juve were criticised after a flat performance in the opening leg and Bonucci claimed Allegri's side were especially motivated as a result.
"Everything that was said hit us on a human level and that brought more out of us," the Italy defender added.
"Our tactical approach was to push down the flanks with Spinazzola and [Joao] Cancelo, keeping their full-backs deep.
"We have a great coach who is leading us into the quarter-finals. We know this is football, people can't wait for you to make a mistake, but the coach and team are big enough to shrug that off.
"It wasn't easy, but we completed the comeback. It's a boost of adrenaline that will push us on to [final date] June 1."