England face Italy in Kiev's Olympic Stadium with a semi-final against Germany the prize for the winners, and Gerrard believes Hart could play a crucial role in what is likely to be an even contest.
England have not reached the semi-final of a World Cup or European Championship since 1996 and have lost three quarter-finals in recent years at the 2002 World Cup, Euro 2004 and the 2006 World Cup.
Gerrard, who played in the 2004 and 2006 defeats, told reporters on Saturday: "I think in Joe Hart we have got one of the best keepers in the world and I think he will become the best in the world.
"His performances so far in the tournament have been fantastic.
"He is one of our key players. We need Joe to perform to the maximum level because at some time in the game tomorrow, maybe during extra time, or penalties, he is going to be called upon to make a fantastic save - and we all trust he can do that."
Hart, 25, has played 21 times for England and is now established as the first choice keeper in a team trying to end a long period without success.
One of the players Hart may have to face is his Manchester City team-mate Mario Balotelli, who came off the bench and scored against Ireland in the group stage last Monday.
"Mario Balotelli is a brilliant player, a very dangerous player and a player we have to take extra care of if he's selected to play," said Gerrard.
"We watch him week-in week-out in the premiership scoring goals and getting assists, we are aware of all his skills and a lot of our players know him very well and hopefully we can contain him tomorrow night."
Balotelli, talking to the media minutes later added: "I think its going to be fun playing against my team mates. They know me and I know them so I hope they are going to be nice. They may be my team mates, but I hope they are going to lose."
Gerrard added that the England players did not need any motivation to lift their game against Italy because reaching the knockout stage in a major competition is always special.
"Our level of performance so far in the tournament has given us confidence and belief. In previous tournaments we have gone out and not performed at the level we are capable of.
"But we have got stronger as this tournament has gone on so, hopefully, that will be the same tomorrow night.
"We are all well aware of how big the game is and what is required.
"The atmosphere among the squad has been superb and it is important that all the players seize this opportunity because, from experience, games like this don't come around very often.
"It's important we give everything out there and we show to all our doubters and our critics that we can compete with the top teams."
Roy Hodgson, who has guided England to four wins and a draw in five matches since taking over last month from Italian Fabio Capello, brought a few wry smiles from reporters when he answered a number of questions in fluent Italian.
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