Milan hit the front on the final bend with 150 metres to race, but was unable to hold off Magnier, with Dutchman Dylan Groenewegen (Unibet Rose Rockets) finishing third.
Magnier briefly raised his hand in celebration after the line, but had to wait for the official photo-finish result before finding out he had indeed won two out of three in Bulgaria.
"I was not really sure that I won the stage to be honest," Magnier said.
"I celebrated and then I say, 'oh, I'm not sure.'"
The Giro's final day in Bulgaria took the riders 175 km from Plovdiv to the capital, with a mostly flat ride punctuated by the Borovets climb and ending with a downhill run into Sofia, perfect terrain for the sprinters to come to the fore.
After two crash-marred stages, Sunday was a more relaxed affair, with Spain's Diego Pablo Sevilla once again part of an early breakaway, but just as in the opening two stages, the Spaniard's efforts came to nothing.
The peloton left it late to reel in the trio out in front, with the leaders holding out until inside the final kilometre. Milan went for broke, looking to outfox his rivals, but 22-year-old Magnier proved that his opening day win was no one-off.
"I dreamed about it, and it was a goal to go for the stage again," Magnier said.
"The team did an amazing job again and we had to really be in a good position at one kilometre to go, and this is what we did."
There was no change in the general classification, with stage two winner Guillermo Thomas Silva of Uruguay holding onto the leader's jersey, four seconds ahead of Florian Stark and Egan Bernal.
Britain's Adam Yates, twin brother of last year's winner Simon, did not start stage three following a crash on Saturday which put the UAE-Team Emirates-XRG rider out of overall contention and left him with delayed concussive symptoms.
The Giro arrives in Italy on Tuesday, following Monday's rest day, with stage four's 138-km ride from Catanzaro to Cosenza.