Di Maria scored the only goal as the Bianconeri drew first blood in the opening leg of the Europa League last-16 tie, thumping home a header from Filip Kostic's 53rd-minute cross.
Following his hat-trick against Nantes in the previous round, the World Cup winner has now been directly involved in 10 goals in 2023 (scored seven, assisted three).
The 35-year-old revealed he is in talks with Juve over extending his 12-month contract, which expires at the end of the season, and Locatelli hailed his team-mate's influence on the side.
"Angel is a phenomenon, he's in another category, I think he's really a champion," the midfielder said. "For us, it's an honour to have him with us in training and in the game. We have to be thankful that he is here.
"We are only 1-0 up, today was only the first half, we must play a mature game there [in Freiburg next week]. The 1-0 may be enough, but we have to go there and play a great game.
Although they lead, Juventus will feel their advantage should be greater, having converted just one of their 20 shots on goal during the first leg.
Head coach Massimiliano Allegri has urged his players to demonstrate a greater clinical edge, though he also encouraged Dusan Vlahovic to stay patient with the striker now having gone five matches without finding the net.
"We have to improve in the number of goals scored," Allegri said. "We create enough, the [conversion] percentage is low. We need lucidity when we attack the area, we have to improve.
"I'm happy because physically Dusan is much better, he has to be calm. Maybe on Sunday [against Sampdoria], he will score."
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."
