The Nerazzurri had numerous chances to win the game during a frantic finale, with Mauro Icardi and Diego Milito both going close, but the Zebrette managed to hold on
Walter Mazzarri's side – who suffered just their second home league defeat of the season on Sunday – were again below-par in front of the San Siro faithful.
In a game of few chances, Inter lacked the sharpness to finish their rare opportunities, although they came close when Danilo D'Ambrosio's header – which came after he followed in Esteban Cambiasso's well-saved strike – was cleared off the goalline by an acrobatic Maurizio Domizzi in the dying stages.
Visiting goalkeeper Simone Scuffet also made a terrific save to deny Hernanes on a disappointing evening for the hosts.
A disastrous 2012-13 campaign under Andrea Stramaccioni saw Inter finish ninth last term, meaning that this season has been one without continental commitments.
And Inter – who sit fifth, in the last UEFA Europa League place – will be nervously looking over their shoulder at Parma, who are just one point behind with a game in hand.
Silvan Widmer's early cross tested ex-Udinese goalkeeper Samir Handanovic but Inter, who brought in defenders Juan and Walter Samuel after Sunday's loss to Atalanta, looked solid.
Udinese may have regretted resting top scorer Antonio Di Natale as their attacks rarely bore fruit, but one of the rare opportunities before the break did fall to the visitors.
Bruno Fernandes' 20-yard effort was awkwardly spilled in the 41st minute by Handanovic, who would have been relieved to see Emmanuel Agyemang-Badu steer the rebound into the side netting.
Inter, growing increasingly frustrated at the lack of success their repeated attacks down the wings were having, resorted to long-range attempts – but Fredy Guarin and Rodrigo Palacio saw their shots fly harmlessly over the bar.
Andrea Ranocchia's mishit cross nearly dipped over Scuffet and in as play resumed, while Guarin – who signed a contract extension on Monday keeping him at the club until 2017 – saw his night to forget end in the 54th minute when he was taken off for Ricky Alvarez.
Roberto Pereyra's curling effort forced Handanovic into a save while at the other end, a poor Alvarez cross only increased the frustration of the home crowd.
Inter finally ramped up the pressure after the hour-mark, a pattern that coincided with Alvarez's switch to the left wing, as Hernanes had two good efforts on goal – the second of which saw Scuffet keep out his fierce 20-yard effort with an impressive save.
Samuel and Palacio then wasted free headers, before Mauro Icardi turned over from close range, and the same player had a shot dropped by Scuffet in the closing stages, only for the goalkeeper to recover quickly.
There was still time for one last effort, but D'Ambrosio was denied as Udinese held on for a draw.