Pual Pogba was denied leading the side and Solskajer has insisted it was not a snub to the French World Cup winner.
But the Norwegian trainer revealed Tuanzebe, who has been a skipper in all his youth levels at the club also possesses leadership qualities.
"Axel is a captain in the making, he's a leader – why not give it to young kids?" he said in an interview with talkSPORT.
"We know we can give it to Sergio and we can give it to Paul, that's no problem and they've been captains before.
"But it was Axel and how does he handle it? He was absolutely fine, he enjoyed it.
"It's just a way of telling him that we trust him.
"Some players learned a lot [against Rochdale] and some proved to us that we can give them more responsibility," Solskjaer added.
"Mason again showed with his goalscoring abilities why we regard him so highly.
"He has two great feet and took the penalty with his right, so he must be a nightmare for defenders having him one on one in the box.
"He can go both ways, which is for me fantastic. You can just see he has got quality and he will get his share of games.
"But he is young and we will give him time to grow."
Having defeated Rochdale 5-3 on penalties after a 1-1 draw, United will host Arsenal on Monday for their seventh Premier League game of the season.
