Seven-time champion Milan hasn't played in the competition since the 2013-14 season.
However, Gennaro Gattuso's team came from behind to beat Parma 2-1 and moved into fourth spot in Serie A — the last qualifying berth for the Champions League.
"We'll enjoy the three points, get our energy back and then we'll see game by game where we can get to," Gattuso said. "I advise the players not to look at the table and to enjoy their two days of rest now."
Milan is one point above Lazio, which was held to a 1-1 draw at last-place Chievo Verona. It is also four points below city rival Inter Milan, which drew 2-2 at Roma.
Only two points separated Milan and Parma going into the match at San Siro, but Parma had already beaten Inter at the famous stadium — and was looking to add another chapter to its fairytale-like rise in its first season back in Serie A since going bankrupt three years ago and being forced to restart in Serie D.
Milan dominated but Parma took the lead against the run of play when Roberto Inglese headed in a corner at the near post in the 49th minute.
Parma's lead lasted less than six minutes before Suso's ball was deflected through to Patrick Cutrone and he fired in off the base of the far post.
And Milan's comeback was complete in the 71st when Franck Kessie converted a penalty after Alessandro Bastoni handled Hakan Calhanoglu's cross.
UNDESIRED DRAW
The draw between Inter and Roma did little for either side.
Inter moved level with second-place Napoli ahead of Napoli's visit to Atalanta on Monday. Roma remained in seventh place and at risk of missing out on Europe.
Moments after Roma had protested for a perceived penalty, Keita Balde put Inter ahead late in the first half by redirecting a cross from Danilo D'Ambrosio.
Cengiz Under leveled for Roma with a powerful long-range effort after the break, and Mauro Icardi restored Inter's advantage by heading in a corner.
Aleksandar Kolarov equalized again with a penalty in the 73rd.
STOPPAGE-TIME TURNAROUND
Genoa almost managed to beat Torino despite playing with 10 men for more than an hour but the home side prevailed 2-1.
Genoa's Romulo was sent off in the 28th minute following two yellow cards in three minutes. The second was for a brutal challenge, which saw the studs of his boot connect with the chest of Torino midfielder Soualiho Meite.
Surprisingly Genoa took the lead less than 10 minutes later. Daniel Bessa headed on a corner and Christian Kouame caught the Torino defense napping to tap in from close range.
Torino turned the match around in first-half stoppage time. Genoa failed to clear the ball and Cristian Ansaldi smashed it into the bottom right corner from the edge of the area.
It took the lead seconds later when Andrea Belotti converted a penalty after Iago Falque had been tripped by Sandro.
Torino moved up to sixth while Genoa remained four points above the relegation zone.
OTHER MATCHES
In Verona, Lazio needed a second-half equalizer from Ciro Immobile to cancel out an early goal from Chievo's 39-year-old captain Sergio Pellissier.
Cagliari is just a point above Genoa after recovering to draw 1-1 at Frosinone. Diego Farias canceled out Francesco Cassata's opener. Cagliari and Italy defender Nicolo Barella was sent off late on following a second booking.
Newly promoted Frosinone remained second from bottom in Serie A, five points from safety.
Elsewhere, relegation-threatened Udinese managed to get a point with a 0-0 draw at Sassuolo.