Zaha, 21, was Sir Alex Ferguson's final signing in charge of the club when he arrived at Old Trafford for €15 million last summer, but he played just four times under David Moyes in 2013/14.
Moyes was eventually sacked after United finished seventh in the Premier League and missed out on European football altogether.
With Dutchman Van Gaal now at the helm at Old Trafford, Zaha is relishing the opportunity to emerge from the shadows and stake his claim for regular football in Manchester after a forgettable season.
"To be honest, if you are not the manager's cup of tea you can't do anything about it," Zaha said when asked about Moyes.
"There was nothing I could do. It was definitely not my attitude. The manager had his way of playing and maybe I just didn't suit it.
"There was so much pressure on him, so I can't really blame him. I could have got a better chance but coming to United brings a load of pressure."
Zaha was the only player in the travelling squad not to have played a minute in the United States prior to Tuesday's penalty shoot-out win over Inter.
But the Englishman was given time in the second half at FedEx Field and he showed glimpses of his potential in an impressive performance.
"I played all right and I'm just happy the manager is happy with my performance really," said Zaha. "I just have to work hard and I have just done what he has asked.
"He said if I worked hard I would get my chance. It definitely is [a fresh start].
"All I can do is work hard and see how the season goes. It's a clean slate for me."
Zaha is down the pecking order at United, with strikers Robin van Persie, Wayne Rooney, Danny Welbeck and Javier Hernandez vying for a spot in the starting XI.
However, the former Crystal Palace man is buoyed by Van Gaal's history of promoting youth.
"I have heard that before he came, so I am buzzing to hear that he will give youngsters a chance," he said.
"That's all I really [wanted] since I came here. I just want to be a Manchester United player. Hopefully this season I will get the chance."