Cummins took two key wickets and effected a crucial run-out on Monday as the Australians defeated Sri Lanka by five wickets to register their first win in India and give the skipper breathing space.
"He led from the front. His spell today changed the momentum of the game," said leg-spinner Zampa, who was named man of the match for his four-wicket haul.
Former Australia captain Michael Clarke had claimed that 30-year-old Cummins was to be left out of the team.
Cumming selection under fire
"I heard last night that Pat Cummins is not going to be selected for this game," Clarke told Sky Sports Radio's Big Sports Breakfast in Australia.
"If Pat Cummins is going to get dropped and not captain the team, then you guys have selected the wrong captain.”
Before Monday, Cummins had overseen a six-wicket loss to India, where his team had been bowled out for just 199, and a 134-rout by South Africa, their heaviest defeat at a World Cup.
However, he put his critics in their place in Lucknow, dismissing both Sri Lanka openers Pathum Nissanka (61) and Kusal Perera (78) after they had put on 125 for the first wicket.
He then ran out Dunith Wellalage with a smart piece of fielding.
"That run-out, those little things can turn scores from 260 to 210 and it's game on. So he particularly led from the front," said Zampa after Australia chased down a 210-run target to win.
"We all back each other 100% in this team, so I don't really know what's been said or what people are reading.”
He added: "There's no pressure on Pat. We've got a really good feeling in the group in terms of backing each other, looking after each other.”
On a day when rain, dust storms and strong winds added some much-needed drama to the contest, Zampa said he was relieved that nobody was injured when scaffolding and hoardings crashed from the roof of the Ekana Stadium down onto the seats below.
Only around 3,000 people were in the 50,000-capacity arena at time, drastically reducing the chances of serious injury being caused.
"I've never been a part of that. Never seen anything like that," said Zampa.
"Quite lucky that there was no one harmed hopefully. The metal pole coming down could have seriously hurt someone.”
Monday's win revitalised Australia's bid to reach the semi-finals although they are only eighth in the 10-team table with one victory and two losses.
Next up is a tricky encounter with fourth-placed Pakistan in Bengaluru on Friday.
"We get backed into a corner and guys step up and we get on a roll, then anything can happen," added Zampa.
"Obviously, really big game against Pakistan and if we can put it all together and play a good game there, you go 2 and 2 and the feeling's much better."