Iago Falque got the hosts off to a strong start midway through the first half but Omar El Kaddouri levelled for ninth-placed Torino shortly after the hour mark.
The Rossoblu quickly re-established their lead through Tino Costa, with Andrea Bertolacci, substitute Leonardo Pavoletti and Costa making the points triply safe in the final minutes.
With three games remaining, Gian Piero Gasperini’s Grifone are just one point behind sixth-placed Sampdoria, who hold the final Europa League spot.
Genoa took the initiative following kick-off, with M’Baye Niang putting Falque through for an attempt that Daniele Padelli pushed to safety.
The visitors were soon on the attack, though, Maxi Lopez’s backheel setting up Emiliano Moretti, whose cross found Fabio Quagliarella for a shot that fizzed just over the crossbar.
It was a false dawn for Torino as Genoa soon broke forward in numbers before scoring the opener.
In the 18th minute Falque found himself at the end of a swift move that saw Marco Boriello arrive in the centre of the box for a cool finish.
Edenilson might have scored a quickfire second had Padelli not smothered his long-range effort and the Torino goalkeeper was sprawling again when Andrea Bertolacci picked his spot from outside the area.
With one minute to go before the break, Torino came within a whisker of the equaliser as Facundo Roncaglia’s mistake allowed Quagliarella to run through on goal, but Mattia Perin stood tall and kept the former Juventus striker at bay.
Quagliarella was withdrawn for Josef Martinez early in the second half after suffering an injury, meaning another Torino player would need to step up.
El Kaddouri answered that call by firing home the equaliser in the 61st minute, his pinpoint free-kick beating Perin after Nicolas Burdisso had fouled Bruno Peres in a dangerous position.
Genoa restored their lead only eight minutes later, however, with El Kaddouri playing a part in this goal too as Costa’s free-kick took a heavy deflection off him to fool Padelli.
Torino strived desperately to force the draw but neither substitute Alexander Farnerud nor Lopez could convert decent opportunities before Genoa struck a critical triple blow.
With three minutes remaining, Bertolacci thought he had rubber-stamped the Rossoblu victory with a sweetly-struck finish off his right boot, yet there was still time for Pavoletti to emerge from the bench and make it 4-1 with an angled drive.
Celebrations were ringing around the Marassi and almost drowned out the players’ cheers as Costa converted his brace deep into stoppage time.