United have won 10 of their 13 Premier League games under Solskjaer, losing their first match in that run at Arsenal last weekend, and have reached the Champions League quarter-finals to leave him as the favourite to take over permanently.
"They have the same principles. Ole's team-talks are big on hard work," Carrick, who won five Premier League titles with United as well as the Champions League and Europa League, told the Times newspaper.
"He delivers it his way. That's the club's way, the DNA of Manchester United: an exciting, positive, winning performance, where you take risks, you go for it.
"Work ethic, humility -- the United way is not 'we're better than you'. We just have to work harder. The best teams United had over the years always had an unbelievable work ethic."
Solskjaer helped United pull off a stunning fightback to reach the last eight of the Champions League on away goals after a dramatic 3-1 win at Paris St Germain, despite losing the first leg 2-0.
United take on Barcelona in the quarter-finals next month.
"I can try to play it down all I want, you try to forget about it, move on, but it was one of those nights that in a few months' time... we'll look back and go 'wow, that was special'," Carrick added.
"The buzz of winning was incredible. The dressing room was bouncing. The lads were all singing 'Ole's at the wheel'. Sir Alex and Eric (Cantona) were in there, so the lads started singing Cantona songs.
"Everyone, young kids to grandmas, come up to me in the street and say 'this is great'. Especially after Paris. Paris was massive."
United are in action later on Saturday with an FA Cup quarter-final clash at Wolverhampton Wanderers.
(Reporting by Shrivathsa Sridhar in Bengaluru; Editing by Christian Radnedge)
