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.
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.
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.
There was still time for one last effort, but D'Ambrosio was denied as Udinese held on for a draw.
