Southampton had a penalty shout after 10 minutes after Scienza raced through and appeared to be caught in the box by two closing defenders, but referee Leigh Doughty was not convinced enough to award a spot-kick.
The U’s then took the lead against the run of play as Harwood-Bellis headed a ball clear but only as far as Goodrham, who let the ball bounce before sweetly striking home from outside the box with his right foot and into the net.
Just over five minutes later, Saints drew level as Scienza whipped in a cross from a free-kick on the right and found the head of Harwood-Bellis.
Oxford started the second half with a big chance to take the lead, when Przemyslaw Placheta struck from the edge of the box and forced a brilliant save out of Bazunu.
The U’s should have taken the lead just after the hour mark when Mills beat two men down the left and darted into the box before passing to the onrushing Nik Prelec, but the forward overcommitted on his run and the pass rolled behind him.
Oxford could have wrapped things up sooner than Mills’ late intervention after Siriki Dembele was played through in the box, but their calls for a penalty were waved away.
The Kassam erupted late on, however, as Mills scored his second goal of the season to claim all three points for his side.
