Roy Hodgson's side, who won every game in qualifying, start their campaign on Saturday against Russia in Marseille.
"If we can be on the right end of key moments, we will go far," said Hart.
The Manchester City keeper also said he "fully believes" in England's defensive options, despite criticism over a perceived lack of strength in depth.
"It's very balanced and very comfortable," said Hart.
"We're going to attack as a team and defend as a team. We'll work hard, we are well regimented and we are ready."
Hodgson's 23-man party - the youngest England squad for 58 years - features three central defenders in Gary Cahill, Chris Smalling and John Stones, with defensive midfielder Eric Dier as a back-up option.
Hart, who has 59 caps, was England's first-choice keeper as they exited the 2014 World Cup at the group stage for the first time in 58 years.
But he is adamant England's past displays are irrelevant: "Playing to a decent standard isn't enough, we have to be ruthless at both ends of the pitch."
England are joined by Russia, Wales and Slovakia in Group B.
With three teams potentially reaching the last 16, hopes are high of replicating the run to the semi-finals in 1996, which Hart watched as a nine-year-old.
"That was a fantastic tournament, one I enjoyed watching and I'd love for other nine-year-olds to have memories of 2016," he added.
Goalkeeper Joe Hart says England must be "ruthless" to succeed at Euro 2016, where he hopes to give a new generation of fans their own Euro '96 moment.