In an evenly-matched first half, Diego Milito missed a golden opportunity to give the hosts the lead, but he blasted his shot over when he was under no pressure.
Further chances for Motta and Walter Samuel should have put Inter in front, but both players were wasteful in front of goal.
Despite the home side's dominance, Udinese were a constant threat on the break, with Antonio Di Natale wrongly ruled offside for one attack.
The second half was end-to-end, but it was the visitors who made the breakthrough on 72 minutes when Mauricio Isla fired in after he was put through on goal by Floro Flores.
In a frantic final five minutes, Inter captain Javier Zanetti was sent off for a challenge in the area on Asamoah. However, Julio Cesar managed to palm after Di Natale's thumping penalty.
A minute later, Inter were awarded a penalty of their own when Milito was hacked down in the box by Damiano Ferronetti. Much to the dismay of the home crowd, Giampaolo Pazzini smashed his penalty well over. To add to the drama, Ferronetti was shown a red card for complaining after the kick had been taken.