Mario Balotelli also struck his first European goal for Milan in a flattering 3-0 second-leg win which took them through 4-1 on aggregate in the qualifying tie.
Seven-times European champions Milan, forced into the playoffs after finishing third in Serie A last season, were under huge pressure to reach the group stage and survived several scares from the Dutch side in the first half.
PSV, with an average age of just over 22 and four teenagers in the starting line-up, were unfazed by the imposing San Siro as they enjoyed more possession.
They nearly went ahead in the eighth minute when Tim Matavz's header was clawed away by goalkeeper Christian Abbiati and Milan took the lead with a low 25-metre shot from Boateng one minute later.
Abbiati made another important save to turn away Adam Maher's 30-metre volley and the visitors had two more chances just before halftime as Jetro Willems' looping cross hit the crossbar before Abbiati scrambled away another Matavz effort.
In between, Milan forward Stephan El Shaarawy rifled a shot against the underside of the crossbar before the ball bounced back into play.
Balotelli, Cup-tied from Milan's Champions League matches last season after his move from Manchester City in January, made the game safe in the 55th minute, diverting Philippe Mexes's flick into the roof of the net from close range.
PSV continued to play good football, only for Boateng to power into the penalty area and add a third with an angled shot in the 77th minute.
Carlos Vela scored both goals as Real Sociedad also dominated at home against France's Lyon to replicate their 2-0 away triumph in last week's first leg.
Vela's opening effort came in the 67th minute and was followed by a run from the halfway line on the final whistle, finishing with aplomb to put Sociedad in the group phase for a second time, a decade after their initial appearance.
Viktoria Plzen will also play in the group phase for the second time where a massive boost to their modest budget awaits.
Stanislav Tecl tucked the ball away in the bottom left hand corner after only three minutes to secure a 1-0 win at Maribor of Slovenia and a 4-1 aggregate triumph.
Two goals from Portuguese international Danny against compatriots Pacos Ferreira helped Zenit St Petersburg complete the formality of a tie decided last week when the Russians won 4-1 away in the first leg.
They won the return match 4-2 in late summer sunshine in Russia for a convincing 8-3 aggregate triumph.
Celtic winger James Forrest scored a dramatic stoppage time winner to secure his club a Champions League group stage place.
Celtic, inspired by the fervour of their Glasgow supporters, overturned a two-goal deficit from the first leg to beat Shakhter Karagandy of Kazakhstan 3-0 and secure a group stage spot for the second year in a row.
They overcame early jitters to score through Kris Commons on the stroke of halftime and then leveled the aggregate tally through Giorgios Samaras early in the second half.
The 3-2 aggregate triumph was secured with only seconds remaining at the end of a mazy run through the defence from Anthony Stokes that set up Forrest's goal for a grandstand finish to maintain the tradition of rousing nights of European competition at Celtic Park.