The result, which came courtesy of a Locatelli brace and a third goal added by Ciro Immobile, confirmed Italy's place in the knockout stage of Euro 2020 after just two games played.
Mancini's men dominated the contest at Stadio Olimpico and were rewarded with a second consecutive 3-0 victory, having beaten Turkey by the same scoreline in their opening Group A match.
Mancini hailed his side's performance, telling reporters: "Switzerland are a strong side. We knew it would be a tough game and it was.
"We created several chances and we deserved the victory. We were struggling in the opening minutes, but then we pressed them high and forced them to make mistakes.
"The lads did really well. It wasn't easy; it was their second game in five days and it's so hot."
He picked out his side's opening goal for special praise, having seen Locatelli start the move in midfield before latching onto Berardi's cross to score from close range.
"This is exactly what the lads have to do, what I'm always asking them to do," said Mancini.
"Always believe that it can be the right chance, as Locatelli did by following the action after his pass to Berardi in midfield."
Mancini's opposite number Vladimir Petkovic faces an uphill struggle to progress from Group A, having taken one point from their games against Wales and Italy.
Switzerland face Turkey in Baku in their final group game, and Petkovic said they must remain committed to their task.
"Many things didn't work for us tonight, and plenty worked for Italy. The truth lies in the middle," he said.
"Congratulations to Italy – they play great football, and have for some time.
"Everyone deserves to be disappointed tonight, but tomorrow from the first training session we move on. I spoke to the team and told them there's a match left, and three points could get us to the next round."
Kieffer Moore headed in 16 minutes from time to cancel out Breel Embolo's opener in Saturday's Group A clash at the Baku Olympic Stadium.
Switzerland managed double the number of shots than Wales and had a Mario Gavranovic goal ruled out by VAR late on, but Moore rescued the Dragons a potentially huge point.
Wales have now lost just one of their last 14 competitive games – against Belgium in March – and Bale is pleased with his side's attitude to battle back in the blistering Baku heat.
"Both teams tried to play. It was a little bit tense, but it's not a bad result and we take the positives," Bale told BBC Sport. "We wanted to win the game and had chances, but so did they.
"When you go 1-0 down and you can crumble but we showed a lot of grit and character like always and we showed our class to fight back and get the goal.
"We worked very hard. It's hard in this heat. I'm proud of the boys. After the match we said we need to use this as a springboard.
"We now need to recover and move onto the next game."
That next game for Wales is once again in Baku against Turkey, who were beaten 3-0 against heavyweights Italy on Friday in the opening match of the tournament.
Like captain Bale, Wales' caretaker boss Rob Page believes his side can take confidence from the point gained against Switzerland.
"We wanted a positive start and it feels like a win in the changing room. We had to dig deep at the end and we had a bit of Lady Luck," he told BBC Sport.
"Turkey are a different opposition to what we've faced today, but we will have to be solid and well organised again.
"Italy have quality in abundance and we respect Turkey too. It's all about focusing on that now and we will be preparing for that test now.
"That's the foundations. If we need to dig in we will but we've proved we can play."
Moore's goal was his sixth for Wales in 18 appearances, four of those being scored with his head.
Page recalled the Cardiff City striker to his starting line-up in a big tactical decision that ultimately paid dividends.
"Kieffer has been terrific at international and club level," added Page.
"He's got a great touch and his link-up play is great. He's got all his attributes for a centre-forward and has given us the rewards today."
Switzerland are now unbeaten in their last six European Championship games in normal time, but Vladimir Petkovic's men had enough chances to pick up all three points against Wales.
Goalscorer Embolo, whose six shots was the most ever attempted by a Switzerland player in a single European Championship match, admits the draw is a disappointing result.
"I feel a bit of frustration as we had the chances for the second goal but didn't take them," he said.
"Perhaps we were too passive after the first goal. We deserved the win, but it turned out to be a draw and that's a pity.
"But we have to take the positives, rest up, focus on the next game and bring the good things from this match into the next one."
