The attacker made the most of a rare playing opportunity in the cup competition, getting onto the score sheet as the Frogs registered a 2-0 home triumph at Estadio Ciudad de Valencia.
The goal is his first for the club in a paltry five appearances in all competitions since completing a summer move from Swiss club FC Zurich.
Levante had to wait until the 80th minute of Thursday's game to take the lead, courtesy of Coke.
The Frogs looked headed for 1-0 triumph but Dwamena made it 2-0 in the third minute of added time, having been set up by Ruben Rochina.
Whereas the 23-year-old lasted for the entire duration of the game, compatriot and club mate Emmanuel Boateng was replaced in the 65th minute.
The win takes Paco Lopez's side through to the Round of 16 in the cup competition.
The Nkawkaw-born's limited playing time at club level has had a toll on his international career, having been overlooked by Ghana coach James Kwesi Appiah since a disappointing showing in a 1-0 defeat to Kenya in September.