Spalletti was in bullish mood ahead of Saturday's clash, insisting that his side would not suffer the same mid-season crisis that beset them last term.
But his side failed to live up to expectations at San Siro, with a poor miss from Icardi seemingly summing up a dismal performance.
Icardi turned from zero to hero with 14 minutes remaining, however, as he lifted a delicate finish home from 12 yards after VAR was used to deem that Seko Fofana had handled in the area.
Joao Mario had Inter's first sight of goal, dragging wide from the edge of Udinese's area, before Icardi headed across the face of goal in the 22nd minute.
Icardi should have done better from Sime Vrsaljko's pinpoint cross as Inter toiled for an opener, but Udinese goalkeeper Juan Musso remained untested until he reacted sharply to tip Keita Balde Diao's crisp strike over.
Inter's profligacy nearly proved costly after the interval, Rolando Mandragora drilling over from Fofana's cut-back, with Hidde ter Avest firing wide moments later, though Inter should have been ahead in the 66th minute. Keita's cross was perfect but, with the goal gaping, Icardi failed to keep his header on target.
Fofana's error gave Icardi the chance to make amends, though, and the forward made no mistake with a perfectly executed Panenka to get Inter back on track. Icardi was flagged for offside as he found the net again with a late header, but Udinese never threatened to punish Inter with an equaliser.