Monza plunged Juventus into fresh crisis on Sunday after beating the under pressure Turin giants 1-0 to win their first ever match in Serie A, while Atalanta are provisional league leaders after winning 1-0 at Roma.
Christian Gytkjaer netted a historic goal 15 minutes from the end at the U-Power Stadium to move Monza, owned by former Italian prime minister Silvio Berlusconi, off the bottom of the table and end Juve's unbeaten start to the season.
Monza took their first three points of the campaign in their first match under Raffaele Palladino, who replaced sacked Giovanni Stroppa on Tuesday.
They are still in the relegation zone but a single point behind 17th-placed Verona, who lost 2-0 at Fiorentina.
"Today was incredible and unexpected," said Palladino.
"They showed themselves to be real men... they gave everything, I shouldn't get any of the credit because it was all down to them."
Troubled Juve, who lost Angel Di Maria to a straight red card five minutes before half-time, sit in eighth and 10 points behind Atalanta after winning just two of their opening seven matches.
Before the match Juve CEO Maurizio Arrivabene said it would be "crazy" to sack Allegri, under contract until 2025 on a reported salary of eight million euros a year.
But a poor start in Italy and two defeats in their opening two Champions League matches have put Allegri in the firing line for supporters, criticised not only for bad results but also for the uninspiring football his team is playing.
Allegri has not been helped by an injury crisis which on Sunday denied him five potential starters, and Juve's chances were hit further when Di Maria -- who later said the defeat was his fault -- was sent off for elbowing Armando Izzo.
Juve had created little before the Argentina winger's dismissal, and Monza went closest to opening the scoring after the break with first Dany Mota and then Gytkjaer fluffing good chances from close range.
Patrick Ciurria had given Mota his chance and the Italian clipped a pass for Gytkjaer who swept the ball home, his first goal in the Italian top flight, sending the home fans wild and leaving Juve wondering where to turn next.
- Atalanta reclaim lead -
Atalanta went top of the league after Giorgio Scalvini's beautifully hit low strike 10 minutes before the break at the Stadio Olimpico earned them the three points.
Gian Piero Gasperini's side are a point ahead of Udinese, who beat Inter Milan 3-1 in Sunday's early match to win their fifth match on the bounce.
However they can be joined on 17 points by either champions AC Milan or Napoli, who face off at the San Siro in the weekend's headline match.
"It was a difficult match against a strong Roma side, but we held our shape well and have too many rsiky moments, especially in the second half," said Gasperini.
Atalanta have conceded just three goals but they rode their luck in Rome, with Tammy Abraham missing two golden chances to level at the end of the first half.
Roma, who lost Paulo Dybala to injury in the pre-match warm up, were livid 10 minutes after the break when referee Daniele Chiffi didn't give the hosts a penalty for Caleb Okoli yanking at Nicolo Zaniolo's shorts to the point that his underwear was visible.
Jose Mourinho had to be held back by his coaching staff and was sent off by Chiffi, meaning he will miss Roma's trip to his former club Inter after the international break.
"It was a very clear penalty," said Mourinho to DAZN.
"I need to change what I tell me players, I have to tell them to not to stay on your feet... be a clown like many who dive like they're in a swimming pool do in this league."
Eldor Shomurodov headed wide after sensational play from Zaniolo with three minutes remaining, and when the Uzbekistan captain dragged a low shot wide shortly afterwards Atalanta were ensured of the points.