His previous two senior goals came for Scotland.
Gilmour had been handed his first start of the campaign in place of Stanislav Lobotka after being restricted to three short substitute appearances so far this season.
However, despite his goal, the former Chelsea and Brighton & Hove Albion midfielder found himself replaced with the Slovak after 58 minutes with the sides level at 1-1.
The narrow win means Napoli lead the table by two points from Juventus after four games.
Meanwhile, Real Madrid’s Weir will return to her home city of Dunfermline on 28 October when Scotland host Switzerland in a friendly at East End Park.
The 30-year-old was the first Scottish woman to be nominated for the Ballon D’or.
