The Nerazzurri's domestic struggles continue as they failed to get a good result against Gian Piero Gasperini's men.
The defender rose to meet a well-worked corner with seven minutes remaining, ending a run of 16 competive matches for Genoa without a win over Inter. Walter Mazzarri's men were frustrated by a tenacious performance from the hosts, and saw a number of chances go begging at a sodden Stadio Comunale Luigi Ferraris.
Inter remains winless in 2014, having lost three and drawn one since the turn of the year, scoring just once in the process.
Gasperini, who came into this game having never previously beaten his former team, made six alterations to the Genoa side thumped 4-0 by Roma last weekend, including a return for ex-Milan striker Alberto Gilardino following his suspension.
Former Genoa hitman Diego Milito was the only new face for the visitors as they sought to improve on their recent poor form in front of goal.
Despite the difficult conditions, the visitors made a high-tempo start, with Ricardo Alvarez racing behind Genoa's defense before unleashing a low drive that was smartly saved by Mattia Perin at the near post.
Perin was called into action again to keep out a stinging effort from Jonathan after 10 minutes, as Inter pressed for an early opening. The Brazilian wasted another good opportunity just before the 20-minute mark, lashing Milito's clipped ball inside the box across goal and wide.
The visitors were dealt a blow when Alvarez limped off injured, but should have been ahead soon after through Rodrigo Palacio, who robbed the ball from Luca Antonini before bursting into the box and firing wide.
Genoa did manage a couple of flashes at goal before the half-hour mark.
Andrea Bertolacci whistled a speculative half-volley just wide, before Stefano Sturaro saw his goalbound effort slowed by a puddle, leaving Samir Handanovic with a simple save.
The Slovenian was tested again five minutes before the break, palming away Giovanni Marchese’s close-range header after Inter struggled to deal with a well-worked corner.
Inter should have taken the lead two minutes after the restart. Water inside the Genoa penalty area helped the ball fall kindly to Palacio, only for the Argentinian to blast over from 8 yards.
The home side responded with chances of its own, with Gilardino failing to get a shot away after being put clean through and Ioannis Fetfatzidis forcing Handanovic to make a sprawling save with a powerful flick-on.
The Greek was denied again just after the hour mark, cutting in from the right before unleashing a powerful dipping effort that was superbly saved. Substitute Fredy Guarin almost provided Inter with the breakthrough when he produced a volley that was well kept out by Perrin.
Palacio then saw a deflected effort saved before Inter was made to pay for their wastefulness as Antonelli rose highest from a set piece to beat Handanovic.
The visitors nearly grabbed a late equalizer through Ruben Botta, whose bouncing header was beaten away by Perrin as Genoa held on for the victory.