Mature Spurs win in Milan
England striker Crouch struck 10 minutes from time following a swift counter-attack at a rain-soaked San Siro, extending Milan's poor recent home record in European competition to just one win in 10 games.
Serie A leaders Milan, seven-times European champions, struggled to impose themselves in a niggly encounter that flared up at various stages.
Milan midfielder Mathieu Flamini was lucky just to be booked for a crude two-footed lunge that forced Vedran Corluka out of the game and Gennaro Gattuso is likely to face UEFA repercussions after appearing to headbutt Spurs assistant manager Joe Jordan after the final whistle.
Gattuso had to be pulled away from Jordan and Spurs manager Harry Redknapp branded the Italian firebrand "crazy."
"I lost control," Gattuso told reporters. "There is no excuse for what I have done. I take my responsibilities for that. I was nervous. I didn't want to argue with players and I did it with him, but I was wrong to do what I have done."
Redknapp also blasted Flamini, calling his challenge on Corluka a "horrendous."
"It was a deliberate tackle. He could have broken his leg. A referee should have red-carded him," an angry Redknapp told a news conference.
Crouch, who finished with aplomb from 10 metres after a lightning break from Aaron Lennon, said Spurs were the better side.
"I thought we deserved to win, we limited them to a few chances... defended magnificently.
"It was a great win, it's a fantastic achievement to beat Milan in the San Siro but the job is not finished."
The Premier League side, who lost 4-3 at the San Siro to Milan's city rivals Inter in the group stage in October, were the better side in the first half before Milan improved after the break.
Spurs had an early shout for a penalty turned down when the fit-again Rafael van der Vaart's effort appeared to strike Alessandro Nesta on the arm.
The home side lost goalkeeper Christian Abbiati after 18 minutes with a head injury. Abbiati went to hospital for checks on concussion and his replacement Marco Amelia had to touch a fierce Van der Vaart shot over the bar after Zlatan Ibrahimovic had fired a free kick wide in Milan's first meaningful chance.
Milan boss Massimiliano Allegri, with several midfielders such as Andrea Pirlo injured and the likes of Antonio Cassano ineligible, sent out a defensive-looking team with centre-back Thiago Silva in midfield and in the first half it showed.
He threw on Brazil forward Alexandre Pato at the break and Milan's Mario Yepes had two headers well-saved by Heurelho Gomes.
Robinho had a close-range shot deflected wide and Milan thought they had equalised late in stoppage-time but Ibrahimovic's spectacular overhead strike was ruled out for a push.