Genoa stayed 17th in the 20-team table, one place above the relegation zone, but are only one point clear of Lecce with one game each to play.
Genoa, Italy's oldest professional club, host Verona in their final game on Sunday while Lecce are at home to Parma. Brescia and SPAL have already been relegated.
Free-scoring Sassuolo, who were always going to be difficult opponents for Genoa, raced to a two-goal halftime lead when Hamed Traore scored from close range and Domenico Berardi curled in a second.
Francesco Caputo helped himself to a brace after the break while Giacomo Raspadori scored the other goal, one minute after coming on.
Jose Mourinho plots cheeky bargain deal for Inter 'outcast' - Euro Papers
Lecce, who have a worse head-to-head record than Genoa, still needed to win at Udinese to earn a stay of execution.
It started badly when Lecce's defence, the worst in Serie A with 81 goals conceded, allowed Samir a free header to put Udinese in front in the 36th minute.
Yet four minutes later, Udinese defender William Troost-Ekong handled in the area and Marco Mancosu, who has missed his last two penalties for Lecce, held his nerve to score.
Lecce kept battling, created some half chances in the second half and were rewarded in the 81st minute when Gianluca Lapadula exchanged passes with Antonin Barak in the penalty area and scored from close range.