It was a far from ideal start to the second half for Cliftonville with the home side awarded a penalty when Keaney was adjudged to have denied Pablo Rodriguez a goalscoring opportunity and Keaney given his marching orders for the foul on 54 minutes.
Rey took the resulting spot-kick and fired into the bottom right-hand corner after Ridd had dived the wrong way.
St Joseph's tried to make the extra man count with Manuel Sanchez's header crashing off the crossbar and Juanma's stunning effort from 30 yards striking the left-hand post.
In the 77th minute, Cliftonville were awarded a spot-kick of their own after substitute Eric McWoods shot struck the arm of Ethan Jolley with Addis coolly slotting home to give ten-man visitors the lead once again.
However, Abraham Paz's side continued to push with Rey's free-kick punched away by Ridd as the striker looked for his second goal of the evening.
And Rey would double his tally and level the game once again on 83 minutes when he headed home Leon Volz's deflected cross.
St Joseph's continued to push until the final whistle, but the Reds defended resolutely to set up an exciting second leg at Solitude next week.