Christian Luyindama got the hosts ahead in the eighth minute but Awoniyi’s excellent solo effort pulled both sides level ten minutes later.
After the break, the striker gave a deft back-heel pass to assist Dorin Rotariu who brought the visitors back on par after Sebastien Pocognoli had restored the lead.
In the 76th minute, Awoniyi completed his brace with a brilliant strike to put his side ahead for the first time in the tie.
But just when the striker thought he had won the game for his side, Mehdi Carcela-Gonzalez and Duje Cop goals in the 87th and 89th minutes secured a comeback victory for the hosts.
Both sides finished the tie a man down after Luyindama and Mouscron’s Omar Govea were cautioned twice.
Awoniyi has now scored six times in 24 Belgian top-flight encounters this season and he will be hoping to replicate this feat when his 12th-placed side welcome Waasland-Beveren to Stade Le Canonnier on February 18.