Harris gave the visitors the lead on 12 minutes after winning the ball high up the pitch, racing through on goal to slide the ball home low to the right of keeper Viljami Sinisalo.
Eight minutes later, Sinisalo saved Elliott Moore’s header on the line, but with half-time looming, Brannagan doubled the lead from the penalty spot, finding the bottom left corner after Josh Murphy was brought down in the area.
But seven minutes after the break, Exeter halved the deficit through Alli, who powered home Tom Carroll’s corner, and that changed the momentum.
Oxford began to retreat, knowing they needed to win, and Moore was required to make a crucial intervention after Alli drove down the left and flashed the ball across goal, but they held on to keep their Championship dream alive.
It looked like Oxford would meet Bolton in the play-off semi-finals but a late Northampton equaliser at Barnsley meant they finished above the South Yorkshire side in fifth place.