The home team earned a place in the group stage 2-0 on aggregate.
Rivers United held the Burkinabes to a 0-0 draw in the first leg two weeks ago.
The hosts missed two penalties in the keenly contested encounter.
Alex Oyowah opened scoring for the Port Harcourt club a minute from time.
Oyowah converted from the penalty spot deep into stoppage time.
The Pride of Rivers are the only Nigerian club, who made it to the group stage of a continental competition this year.
They made it to the quarter-finals stage of the competition last season.
Stanley Egumas side will look to make history as the first Nigerian club to win the CAF Confederation Cup.