Guido Marilungo almost brought his side level just minutes later, but he could only direct the ball over the bar from close range.
In the final minute of the first half Palacio had the ball in the back of the net, but his strike into the far corner was correctly ruled out for offside.
In the second half Hernanes came close to a second for Inter when he fired just wide of the post following a powerful run.
Mateo Kovacic was the next to threaten for the visitors with a stinging drive from the edge of the box that whistled just wide.
With five minutes of the match left Inter thought that they had doubled their lead when Hugo Campagnaro touched in Hernanes's free kick, but the defender was adjudged to have been standing in an offside position.
Icardi and Palacio then both saw efforts saved in stoppage time as Inter held on to move back into the top half of the table.