Bonucci struck after Dani Alves had given the holders the lead to secure a record third consecutive triumph in the competition and keep their hopes of a treble of league, cup and Champions League alive.
Juve went into the match at Stadio Olimpico on the back of a 3-1 defeat to Roma on Sunday – a result that denied them the chance to seal the Serie A title.
Bonucci was at a loss to explain that poor performance but felt it served them well as they approached the Lazio showdown with far greater conviction.
"In the second half against Roma, we were sluggish and not determined enough. We gave ourselves a good talking-to," the centre-back told Rai Sport.
"We knew before that this was the real Juventus. It was a shock when we didn't approach Roma in the right way but it was a wake-up call we needed."
Neto kept his place in goal ahead of Gianluigi Buffon and made a handful of good saves early in the second half to deny Ciro Immobile and Sergej Milinkovic-Savic.
"Lazio have a lot of quality in attack, so it was natural that in a one-off game we'd let them have chances, but fortunately Neto rescued us," Bonucci said.
"The Coppa Italia is thanks to everyone: it allows those who didn't play as much to make their mark and I thank the coach and club for giving the team so much quality and tenacity this season."
Juve can clinch the Scudetto against Crotone on Sunday with a win, although the title will be theirs regardless if Roma lose to Chievo the day before.
Bonucci wants to clinch the domestic crown this week in order to give Juve ample time to prepare for the Champions League final with Real Madrid on June 3.
"The match was tough but we approached it in the right way. Now we have to close out the Scudetto as soon as possible," he said.
"Sunday is another final, a chance to take home the second objective of the season, and then it's on to Cardiff."