England suffered a penalty shoot-out loss at Wembley against Roberto Mancini's side, with Saka missing the vital spot-kick.
Keane was quick to criticise experienced performers Grealish and Sterling, as he suggested they should have taken responsibility ahead of Arsenal youngster Saka.
"If you're Sterling or Grealish, you cannot sit there and let a young kid walk up ahead of you," he told ITV Sport.
"You can't sit there, see a young kid at 19 who is shy walking up in front of me, and I've played a lot more games and have a lot more experience?
"Sterling has won trophies. I'm not saying he wasn't prepared, maybe Gareth had him sixth or seventh.
"You can't sit there, that must be hard to take. You pull this young kid and say 'listen, I'll step up for you."
Despite his doubts over England's spot kick takers, former Ireland and Manchester United captain Keane insisted Gareth Southgate's side can take some encouragement from their performance in the tournament, with the next World Cup a little over a year away.
"It was an amazing night, a brilliant night of football," he added. "I didn't think either team deserved to lose. Someone is going to miss a penalty, but this is not a night to criticise Gareth or the players who missed. Instead of criticising the players who missed penalties, I admire the players who stepped up.
"Both keepers made some brilliant saves and this is a tough night for England. They scored too early. Credit to Italy, they showed great belief, but England will learn from this. Sometimes you have to lose to eventually win. The World Cup is around the corner, and they have to grow from this.
"For all their plans in penalties, it didn't work for them. With a huge goalkeeper in front of you, it's not easy. As Mike Tyson says, everyone has a plan until they are punched in the face. It's not easy in a penalty shoot-out.
"It is all about players learning and maturing now. These players need a break now. They have done well, but it is tough on them."
Former England defender Gary Neville insisted England could hold their heads high after a painful defeat at their Wembley home.
"The lads fell a little bit short but they should be so proud," said Neville. "Every single one of them has been outstanding through this tournament.
"We'll have to look after the young lads (who missed penalties). They will need a cuddle. The whole nation is going to need cuddling.
"It was a massive struggle but it was always going to be a massive struggle. It was 50-50. Gareth has planned for penalties. He brought his subs on ready to go."
On Rashford and Sancho missing penalties after their late introductions, former England striker Alan Shearer said: "It's a big ask to put players on with a minute to go and say 'go and take a penalty' when they've had no feel of the ball. You're put in a position."
Rio Ferdinand gave his verdict on England's heartache as he stated: "That's happened for years, the last five, four, three minutes, players go on as takers.
"Marcus Rashford is a recognised taker. It happens. We look down the years, the biggest and best players miss penalties. Big players of the past have missed penalties.
"Huge disappointment, but the fans clapping our players, they brought us great joy that my generation and below have not felt. They lifted this country when we needed it after a pandemic and take credit for that.
"It's harsh, painful. I was supremely confident, but they are wily old foxes in that (Italy) team. When it got to penalties, it's a toss of a coin."
Former England midfielder Frank Lampard added: "In respect to Italy, they controlled the elements of the game after 20 minutes and you have to give them credit. We wanted our team to get over the line, but I do think Italy were the better team."