Roberto Mancini's men found the back of the net twice through Mauro Icardi and Rodrigo Palacio but were unable to come away with a result
Juraj Kucka headed home a dramatic late winner to decide a pulsating encounter, which leaves Roberto Mancini's men needing a miracle to qualify for the Europa League.
Earlier this week it was confirmed that Genoa - who started the game in sixth - would not be granted a Uefa licence to play in Europe next season, meaning that even if they maintain their current standing the team finishing seventh will play in Europe's secondary competition.
But Inter twice let the lead slip in a pulsating encounter at Luigi Ferraris, meaning they are two points behind seventh-placed Sampdoria - who play Empoli on Sunday - with just one game to play.
In an action-packed first half Mauro Icardi prodded Inter ahead just before the 20-minute mark, only for Leonardo Pavoletti to restore parity.
Rodrigo Palacio finished a swift breakaway to put Inter back in front, but again Genoa responded when Maxime Lestienne - who had earlier rattled the crossbar - capitalised on a defensive mix-up.
The second half was just as exciting and the woodwork denied Hernanes and Marcelo Brozovic a winner for Inter in the same attack, and there was worse to come when Kucka netted in the 89th minute as Genoa climbed to fifth.
In a high-octane start Samir Handanovic kept the scores level with a stunning save when he flung himself to the right to keep out Lestienne's point-blank header on the goalline.
Having survived that scare, Inter made the breakthrough in the 19th minute. Icardi's header from Hernanes' cross hit Edenilson on the back and the striker pounced quickest to prod past Mattia Perin.
The lead lasted just five minutes, though, as Pavoletti turned Andrea Ranocchia on the edge of the box and rifled an unstoppable half-volley past Handanovic.
Inter were back in front on the half-hour mark as Palacio collected Icardi's flick and calmly slotted past Perin when one-on-one to finish a lightening counter-attack.
The pulsating action continued as Pavoletti crashed a fierce volley against the crossbar, before Icardi twice had the ball in the back of the net only to be rightfully denied for offside on both occasions.
Genoa made the visitors pay with an equaliser four minutes before half-time as Handanovic and Ranocchia got their wires crossed when dealing with a throughball and Lestienne slotted into the empty net.
The hosts piled forward at the start of the second half and Pavoletti had claims for a penalty turned down when he went to ground under a challenge from Juan Jesus.
Handanovic made another fine stop to deny Andrea Bertolacci's long-range effort, before Nicolas Burdisso headed narrowly wide of the right-hand post from Lestienne's corner with under 20 minutes remaining.
Inter were then visibly frustrated as Hernanes' low drive rebounded off the left-hand post and Brozovic curled the follow-up against the bar.
Hernanes and Icardi both missed chances as Inter pushed for winner, and Kucka dealt the sucker punch when he leapt highest to meet Edenilson's free-kick a minute from time.
There was almost one last twist when Icardi claimed for a penalty in injury-time when he went tumbling from Burdisso's challenge, but referee Paolo Tagliavento was uninterested.