Two of the defender's former clubs will battle it out for a Scottish Cup final berth when Jackie McNamara's side face Rangers.
Youngsters Andy Robertson, John Souttar and Ryan Gauld have been the shining lights of a decent season for United so far.
And former Scotland star Dailly said: "The way I see them and the way Jackie McNamara seems to have looked after them, these big stages almost tend to be fun for them.
"They are allowed to go and express themselves and it looks like it's a bit of fun for them.
"Sometimes the game can be so big that nerves go out the window. Sometimes league matches can be more nerve-racking and it feels a bit more tense, and you need a result and feel a bit tired.
"These boys will be fully rested and fully ready. If any young lads don't play well this weekend, it's probably because Rangers have done their homework and manage to stop them."
Dailly was part of the United side that shocked Rangers to lift the Scottish Cup in 1994 under eccentric boss Ivan Golac.
And the 40-year-old, speaking on behalf of TSB, sponsor of the Scotland Under-18 Schoolboys team, who face England in the Centenary Shield on Friday, said: "We were just resting up and he told us to go and express ourselves.
"To do that, you have got to be relaxed and confident that your manager is allowing that, to go out and have nothing to lose. It definitely had a big effect on us.
"They have got more structure under Jackie in the positions they play and the way they are coached at the club in general, the whole thing seems set up to allow them to do go out and use their ability."