The Rossoneri were hovering above the relegation zone after losing five of their first eight league games but, having received the backing of the club's board, Allegri steadily steered his side into European contention.
Indeed, a fine run including just one defeat in 2013 left Milan best placed to take third spot on the final day, a mission they completed with a come-from-behind 2-1 victory at Siena authored by a Mario Balotelli penalty and Philippe Mexes' late strike.
Allegri told acmilan.com: "In the 20-odd years of the (Silvio) Berlusconi presidency third place has always been the minimum objective set down for Milan.
"After a disastrous start to the season the team put in a marvellous second half and only lost one match so I have to thank the players for what they have done.
"I am very pleased with them and they had to battle hard. It was an important win which means we are in the preliminary round of the Champions League.
"The club stuck by me in the difficult times so I have to thank them and I have to thank the fans who always got behind the team."
Claudio Terzi's opener in the 25th minute left Milan fans listening nervously to events at Pescara, where fourth-placed Fiorentina were cruising to a 5-1 win.
The pressure was really on when midfielder Massimo Ambrosini was sent off for a second bookable offence but Christian Terlizzi swiftly followed him down the tunnel to even up the teams, before Balotelli levelled with an 84th-minute spot-kick.
And three minutes later, Mexes was on hand to bundle home a rather unattractive winner.
Reports in Italy have suggested Allegri could now opt to end his three-year association with the San Siro outfit in pursuit of a new role, but vice-president Adriano Galliani is keen to retain his services.
He said: "We have been to hell and back and six minutes from time we were out of the Champions League. It looked all over when Ambrosini was sent off but no one takes a penalty like Balotelli.
"Over the last three years Allegri has earned us more points than Juventus. On net points, we are on top and hopefully we will continue with him. We will sit down and talk."
Siena, meanwhile, head into Serie B after collecting only 30 points, eight short of 17th-placed Genoa's tally, meaning the six points deducted for match-fixing at the start of the campaign were not a decisive factor.
Coach Giuseppe Iachini told acsiena.it: "There is a great deal of regret after this defeat, we certainly deserved to leave the pitch with a positive result.
"The team fought with pride, as they should have, but we were unlucky with certain incidents.
"We finished with professionalism and in the end other teams' rankings would not have affected us in any way."