The visitors took the lead after 77 minutes at Hogwood, when substitute Jonathan Afolabi was fouled and Sam McQueen converted the penalty.
Just two minutes later, Armani Little was sent off after receiving a second yellow card for a foul on Josh Barrett.
Reading piled on the pressure and in the 90th minute were rewarded when House collected the ball and fired low into the bottom corner.
That result means Reading are third, three points behind leaders Middlesbrough, while Southampton are up to eighth on 14 points.
Reading: 8 Jan v Brighton (H), PL2
Southampton: 8 Jan v Blackburn (A), PL2