The Young Lions have not tasted success in the competition for nearly 30 years, having won the trophy in both 1982 and 1984.
Stuart Pearce's side are among the favourites for this year's crown in Israel and have been indomitable of late.
Thursday night's 3-0 defeat of Romania means they have not tasted defeat since November 2011 and won eight on the trot without conceding a goal - facts not lost on Zaha.
"I think England have a massive chance, to be honest," the attacker said.
"The strikers can just focus on scoring goals because we know at the back there are no goals being conceded.
"It lets us work on what we're doing up top.
"The team is bonding well together and playing well together.
"We're looking forward to the tournament and with the team we've got I am sure we'll get something out there."
Zaha was England's outstanding player on Thursday night at Wycombe's Adams Park, netting an exquisite first goal for the under-21s before substitutes Jack Robinson and Nathan Delfouneso added extra gloss to the scoreline.
"It was a good game, to be honest," Zaha said. "The whole team were at it.
"It was just enjoyable to be part of it. I am happy to finally get a goal because there have been a couple of games I've been trying to get on there.
"I can't even explain how it feels. After all the hard work I've done, I was happy to get that goal."
Zaha was deployed as lone frontman against Romania and put in a far more assured performance in the role than last time out against Sweden.
"It is good fun to play in," he added.
"Everyone is on my wavelength, really, with little one-twos and things like that.
"It was good, just knowing the players I am playing with at the top and are talented.
"Every run I make they will play me through and made it easier for me playing up there."
Thursday night was the Young Lions' penultimate game on home soil ahead of the European Championships.
The last is Monday's clash with Austria in Brighton, although star man Zaha will not be available for it.
The 20-year-old, who is seeing out the season with Crystal Palace before linking up with Manchester United, has returned to his club to have dental work already scheduled for Monday.
Henri Lansbury has also left the squad, returning to Nottingham Forest having been forced off with an injury after two minutes last night.
Goalkeeper Declan Rudd is another to have left the squad as he has reported back to Preston, where he is on loan from Norwich, so he can feature against Leyton Orient on Saturday.
Manager Pearce has, though, added one player to his squad in the form of Leicester defender Michael Keane. The 20-year-old Manchester United loanee will link up with the Young Lions on Saturday.