Antonelli headed home in the 83rd minute to drop Inter further off the pace in the hunt for Champions League qualification.
They have still not won since before Christmas and have scored only one goal in their four games this year.
The hosts started brightly, with Alberto Gilardino blasting a free-kick over the top, before Inter grew into the match.
Ricky Alvarez tested Mattia Perin with a low shot from inside the area, while the keeper also had to be alert to stop Jonathan Moreira's shot at his near post.
Zdravko Kuzmanovic was the next to have a go for Inter, but his effort went went, with Moreira firing another attempt high.
Genoa went closest to taking the lead in the 26th minute as Andrea Bertolacci's right-footed shot from outside the area flew inches past the left post.
Stefano Sturaro then worked Inter keeper Samir Handanovic, although in truth his shot from the centre of the area was easily saved.
Handanovic kept out a long-range Bertolacci shot early in the second half, but Inter were still creating chances.
Fredy Guarin's shot was headed for the bottom corner before Perin got down to make the save, while Diego Milito's shot flew wide.
But Antonelli made the telling contribution in the 83rd minute.
He rose to head home following Bertolacci's corner.
And the left-back went close to a second four minutes later, shooting wide from the centre of the area, as Inter were left empty-handed again.