That early goal didn't set Vanuatu on course for a big win, however, and the Cooks equalised through Jeremiah Williams on the stroke of half-time.
Vanuatu scored a second from Jimmy Moso shortly after the restart and the scored remained 2-1 to Vanuatu, who now progress to the quarter-finals.
Both teams had lost to Tahiti in their opening games.
In the day's other pool match, a dominant second half saw New Caledonia beat American Samoa 7-0.
The final group match takes place on Wednesday.