Solskjaer, last month appointed as the permanent United boss, famously scored the injury-time winner in the Champions League final when Alex Ferguson's United side won three trophies in 1999.
The Norwegian, preparing his side to face Barcelona in the first leg of their Champions League quarter-final at home on Wednesday, intends to build the side around the principles that worked so well 20 years ago.
"We challenged each other every day in training," he told ESPN. "There were fights and there was an inner justice that I've not seen in too many teams.
"We never, ever accepted anyone being below par in training. We kicked each other and we had to win every single day. That moved over into games and we never gave in. There was an unbelievable team spirit.
"I want my players to really push each other and demand 100 percent attitude from each other every single day. That's the only way you can improve and it's the only way you can win.
"We had something special. It's a different group now and the challenge is, 'How can I make these players winners as we were?'"
United are currently sixth in the Premier League but are just three points behind third-placed Tottenham and are still in contention in the Champions League.
United are the underdogs for the Barcelona tie but Solskjaer, who oversaw a stunning comeback victory over Paris Saint-Germain in the last round, is confident.
"We're playing against a good team but there is something special about this group of players," he said. "It will be fantastic."
And the United boss is not worried that Manchester City could surpass United's achievement in 1999 by winning a quadruple this season.
"United fans want us to win it (the Champions League) because they want to win it," Solskjaer said. "I think we should look at what we can achieve and not what we can stop others achieving."