Bastoni was brought on with 10 minutes to go and headed in a rare goal in the fourth minute of time added on.
Empoli were one off the bottom of Serie A when Nicola replaced Aurelio Andreazzoli as coach last month - they have not lost since, winning three of six to move up to 13th, and are now five points above the relegation zone.
Nicola has a reputation as an escape specialist, having helped Crotone and Salernitana avoid relegation in recent seasons.
Sassuolo, in contrast, are in freefall. They have lost five of their past six matches to sit level on points with 18th-placed Hellas Verona.
Empoli took the lead in the 11th minute when Sebastiano Luperto headed in Razvan Marin's free kick.
Sassuolo were handed a way back into the match nine minutes into the second half when they were awarded a penalty for a foul on Ruan by Ardian Ismajli.
Andrea Pinamonti had missed from the spot against Atalanta last week but he made no mistake this time.
Empoli restored their lead with a penalty of their own 10 minutes later, M'Baye Niang winning and converting the spot kick after his shot was handled by Sassuolo captain Gian Marco Ferrari.
Ferrari redeemed himself 13 minutes from time when he headed Nedim Bajrami's free kick into the bottom left corner to level things once more.
But Bastoni was to have the last word, scoring his first goal in nearly two years to snatch a vital win for Empoli.
Daniel Maldini scored his first goal for Monza to set his new team on the way to a 2-0 win at bottom club Salernitana.
Maldini, who joined on loan from AC Milan in January, played a perfectly timed one-two with Roberto Gagliardini before giving Monza the lead in the 78th minute.
The 22-year-old is the son of Paolo Maldini and the grandson of Cesare Maldini - both former Milan captains.
Monza skipper Matteo Pessina sealed the match five minutes later with a delicate chip over Salernitana goalkeeper Guillermo Ochoa.
Mateo Retegui scored a stunning overhead kick as Genoa beat 10-man Udinese 2-0.
A cross was deflected into Retegui's path by Udinese defender Lautaro Giannetti and the Italy forward gave Genoa the lead in acrobatic fashion in the 36th minute.
Mattia Bani headed in Genoa's second four minutes later.
Udinese defender Thomas Kristensen was sent off four minutes into the second half after receiving a yellow card either side of the interval.
