Police discovered vomit in the vehicle and Lloris had to be helped out.
"He is the first to regret about all that happened", Pochettino told reporters at the club's training ground ahead of Saturday's home clash with Liverpool.
Spurs plan to fine the France star £250,000 - two weeks' wages - following his court hearing on Wednesday.
Alli, meanwhile, picked up a hamstring duty while on global duty for England during their 2-1 UEFA Nations League defeat to Spain.
"Of course he doesn't feel great". He said he was guilty and responsible. I made a mistake.
"Of course he's punishing himself".
"We are not God, we can't judge people". He understands that it was a mistake. It is a massive lesson for everyone. "Now I need to pay like a man".
"I wish to apologise wholeheartedly to my family, the club, my team-mates, the manager and all of the supporters. It's not a moment to take risks now". Then [we must accept] we are men and women and of course we can... He was lucky there was no outcome. That is important. And of course now it is a massive lesson for him, a life lesson. "Because he is clever, he learns and now he is going to move on". But he has the capacity to move on and to learn and be focused on playing football.
The club also felt he played while carrying an injury at times during the World Cup this summer.
It seems likely both Spurs' captain and Alli will miss Tuesday's Champions League Group B match at Inter Milan too.
Pochettino insists he is not anxious by Harry Kane's performances so far this season.
He denied the forward needs a rest, adding: "We know that it's not a situation that worries us". There's no doubt that he's one of the best strikers or players in England and in Europe.