AC Milan forward Stephan El Shaarawy, celebrates after scoring during a Serie A match between AC Milan and Udinese
The result left Milan just a point above the relegation zone and means that neither of the two Milan giants have a victory after three games of the new season.
Inter Milan's 3-1 defeat at Novara yesterday, which led to coach Gian Piero Gasperini being sacked and replaced with Claudio Ranieri, means they are in the bottom three, a point behind Milan.
Genoa are the early Serie A leaders on goals scored beating Catania 3-0 today.
In Milan, the champions were hoping to avoid the pitfalls that befell their neighbours the day before but in the end were lucky to escape with a point.
In a quiet start the only clear chance came from an Alexandre Pato header that drifted well off target.
That was just about the Brazilian forward's only contribution before he limped off to be replaced by Italo-Egyptian teenager El Shaarawy, an Italy youth international who spent last season on loan at Serie B Padova.
Milan shot themselves in the foot on 29 minutes as goalkeeper Christian Abbiati spilled a Gabriel Torje cross from the left and Antonio Di Natale accepted the gift to shoot into an empty net.
Marauding full-back Pablo Armero then tested Abbiati's nerves with a rocket the keeper stopped at his near post.
The closest Milan came in the first period was a bullet free-kick from Dutch veteran Clarence Seedorf that beat Samir Handanovic but crashed back off the post.
In first half injury time Antonio Cassano teed up Antonio Nocerino but Handanovic was out quickly to block at his feet before Ignazio Abate fired the rebound wide.
The champions turned up the pressure after the restart and Handanovic had to make another crucial stop to deny Cassano a clear scoring opportunity.
Seedorf clipped the ball over the top and Cassano took it down on his chest before firing against Handanovic's legs from seven metres.
It was backs to the wall for Udinese and they crumbled on 63 minutes as Cassano slipped the ball to his right and El Shaarawy pulled the ball back across a wrong-footed Handanovic and inside the far post.
That sparked Udinese out of their shell and they almost had the lead again 15 minutes from time when Di Natale teed up Giampiero Pinza, who cracked a shot against the inside of the post, the ball rolling across the line before going wide of the far post.
In the closing stages Milan were indebted to Abbiati for holding onto a point.
First he scrambled across his line to claw away a header from Morocco centre-back Mehdi Benatia before moments later getting down quickly to save a snap-shot from Di Natale.
Rodrigo Palacio, a transfer window target for Milan, scored twice before Guinea midfielder Kevin Constant added the third to send Genoa top.
Juventus remain just behind after a 1-1 draw at home to Bologna.
Mirko Vucinic opened the scoring for the hosts before getting sent off before half-time.
Daniele Portanova scored the equaliser seven minutes into the second period.
Napoli dropped off the top after losing 1-0 at Chievo as coach Walter Mazzarri's team selection backfired spectacularly as he left out five first-team regulars.
Atalanta moved off the bottom following a 2-1 win at Lecce that means they even moved above Inter and out of the relegation zone despite starting the season with a six-point penalty.