Monaco had to wait until the seventh league match of the season to record their first win, following three defeats and three draws.
Leonardo Jardim's team took the lead in the 29th minute when Golovin fired the ball into the top corner for his first goal of the season, only for Nice to level the tie just after half-time through Patrick Burner.
Following the equaliser, Monaco retook the lead thanks to a second from the Russian midfielder before Wissam Ben Yedder's strike sealed the three points.
The French forward had been in good form going into the game, with four goals already this season following his summer move from Sevilla, before he added a fifth from just six appearances.
After a quiet start to the game, Monaco took the lead after a clever move from a corner, as the ball was pulled back to 23-year-old Golovin, who expertly placed the ball into the top corner to give his side the lead.
The home side were denied another, when VAR judged there was an offside before Islam Slimani hit the ball into the back of the net.
Monaco continued to put the pressure on the Nice defence towards the end of the second half but were unable to find a second goal.
Nice started the second half brightly and were rewarded for their intensity in the 53rd minute when Burner found himself at the edge of the box after a corner was blocked by the Monaco defence, and he took a touch before smashing the ball into the net to level the scores.
Against the run of play, Monaco took the lead for a second time after a long ball found Slimani, who went around his defender before squaring the ball to Golovin for his second of the evening.
Minutes later, the hosts secured the result without a doubt when goalscorer Golovin turned provider, sliding the ball into the box for substitute Ben Yedder to turn home to make it 3-1.
In the final minutes of the match, Nice continued to attack the Monaco goal and were kept at bay by some strong defensive tackles, reducing Patrick Vieira's men to shots from range.
Monaco squandered an opportunity to extend their lead further in stoppage time, when Ben Yedder sent a curled effort straight at goalkeeper Walter Benitez in the Nice net.