The victory, which avenged a 1-0 loss to Dominica in the 2010 Caribbean Cup, finished the Dominican Republic atop first-round Group 4 with seven points, one more than Barbados, which beat Aruba 2-1 in the second game of a doubleheader.
Fana scored in the 66th and 93rd minutes - the finale coming eight minutes after Dominica's Chad Bertrand was ejected for a second yellow card, sending the Dominican Republic to Group 7 in the second round with host Guadeloupe, Martinique and Puerto Rico October 23-27.
Barbados also kept its hopes alive, getting goals from Barry Skeete and Mario Harte to finish second in Group 4. The best second-place team from either Group 4 or Group 5, which begins play on October 10 in Trinidad & Tobago.
Skeete gave Barbados the lead in the 21st and - after Jean-Luc Bergen equalized with his second goal of the tournament in the 32nd - Harte restored the advantage in the 56th. Aruba finished the match with 10 men after captain Raymond Baten was ejected in the 89th with his second yellow.
Dominica finished third in Group 4 with three points, two ahead of Aruba.
The winners and runners-up of each of the three second-round groups will advance to the Caribbean Cup finals in Antigua & Barbuda along with the host and Jamaica December 3-15.