Pearce's side went down 3-1 in Petah Tivka against Norway on Saturday and the result, following an opening defeat to Italy, means they can no longer qualify from Group A.
But Henderson thinks Pearce, who is out of contract at the end of the month and due to hold talks with the Football Association, should remain in charge.
"Of course [he should stay]. He has been a brilliant manager since I've come into the U21s," said the Liverpool player.
"Obviously you can see his record - we got to the final a few tournaments ago and lost to a good team in Germany.
"He has been brilliant with all of us so I would love to see him stay on."
Meanwhile, the defeat against Norway left the midfielder in a state of shock.
"I can't believe it," added Henderson.
"There was high hopes going into the tournament. It is very disappointing and all the lads were devastated in the dressing room, but I think we battled and kept on going to the end.
"I think the Italy game was really bad. We didn't play well at all and got beat, which was really disappointing.
"I thought we played better against Norway, to be honest. I think we dominated the game, dominated possession of the ball but we have conceded very bad goals as well so we can't believe it, really."
England face hosts Israel in the final game of Group A on Tuesday in Jerusalem.