Inter went into the game with the best home record in Serie A, having won nine, drawn three and lost just one of their 13 preceding games – but it was Genoa who led at half-time through Rodrigo Palacio.
But Giampaolo Pazzini drew the hosts level before Eto’o struck twice, Goran Pandev further extended Inter’s lead and Yuto Nagatomo sealed the deal - while Mauro Boselli grabbed a stoppage-time second for Genoa.
Inter, who cut Milan’s lead at the top to five points, had the ball in the back of the net after 11 minutes but Eto’o was adjudged to have strayed offside after poking Maicon’s diagonal through-ball past Eduardo in the Genoa goal.
Cameroon international Eto’o was at the forefront of every Inter attack in the opening half-hour, and three times he cut in from the right without really testing Eduardo.
But it was Genoa who opened the scoring, with Palacio lashing home an excellent finish into the far corner from the right side of the box.
If Inter boss Leonardo responded angrily in his half-time team talk then it certainly worked as his side were soon level, star of the show Eto'o switching play to the right, with Maicon leathering in a low cross which Pazzini rifled in from close range.
And Inter were ahead just 60 seconds later, with Eduardo spilling Pandev's shot, allowing Eto'o to pounce on the loose ball and poke home.
Eto’o, who signed from Barcelona in July 2009, soon chalked up his second, stealing possession from Giandomenico Mesto on the left edge of the box before driving goalwards and finishing clinically.
Inter had turned utterly dominant and Eto’o sparked the fourth, finding Wesley Sneijder with a clever reverse pass, before the Dutch international squared, allowing Pandev to slot the ball into the net to cap a fine team goal.
That just left Nagatomo, who joined from Cesena in January, to make it five, turning on the penalty spot before firing into the roof of the net, before Boselli scored the game’s seventh goal with a diving header at the back post.
In other matches, third-placed Napoli, beaten last Monday by Milan, suffered a fresh blow when they drew 0-0 at home with second-bottom Brescia to slip eight points behind the leaders with 10 games left.
Udinese are now just three points behind Napoli after they followed up last weekend's 7-0 away thrashing of Palermo with a more modest 1-0 home win over all-but-doomed bottom side Bari thanks to a penalty from Serie A top scorer Antonio Di Natale.
Adrian Mutu, banished by Fiorentina for weeks for demanding a move before he apologised, started to repay the mid-table Florence side with two goals in a 3-0 win over coach Sinisa Mihajlovic's former team Catania.
Diego Simeone's outfit are slipping dangerously close to the drop zone, which Cesena clambered out of with a 3-2 win at a woeful Sampdoria side who still have not come to terms with the January departures of Pazzini and Antonio Cassano.