Saka, 19, had to score to send the shootout to sudden death but his kick was saved by Gianluigi Donnarumma.
The Italian goalkeeper had also saved from Jadon Sancho after Marcus Rashford had hit the post.
Former Manchester United captain Keane said more experienced players should have stepped up and taken the responsibility.
“If you’re Raheem Sterling or Jack Grealish, you cannot sit there and have a young kid (Saka) go up for a penalty ahead of you,” former Ireland international Keane told ITV.
“They have a lot more experience, Sterling has won trophies, they had to get in front of the young kid and stand up.”
AS Roma manager Jose Mourinho echoed Keane’s comments.
“For Saka to have the destiny of a country on his shoulders, I think it is too much for a kid to have everything on his shoulders in this moment,” Mourinho told talkSport.
“But I do not know if I have to ask that question to (coach) Gareth Southgate or not.
“Because many times what happens is that players who should be there, they are not there. Players who should be there, they run away from responsibility.”
England playmaker Grealish said he wanted to take the penalty kick.
“I said I wanted to take one!!!!” he tweeted.
“The gaffer has made so many right decisions through this tournament and he did tonight!
“But I won’t have people say that I didn’t want to take a peno when I said I will.”
Former England defender Gary Neville said the decision on who was taking the penalties would have been made well before Sunday’s final.
“They would have worked out over the last few weeks in camp, done sessions on it, looked at who’s scoring the most and got the best record,” he said.
“It would be scientific, it would be data-led.”
Former England defender Rio Ferdinand said no one could “blame and point the finger” towards the players who missed the penalties, while Alan Shearer said players should be praised for stepping up.
“For the youngsters to say, ‘yes, I’ll go on and take a penalty’… you have to give them credit for that,” former England skipper Shearer told the BBC.
“But it will be tough for them now. Hopefully, they will get over it.”