Mane finished from less than a yard out with five minutes left, cancelling out a majestic Christian Benteke header that appeared to put Liverpool on course for three points.
Liverpool have drawn all three games under Klopp, and a share of the points was a fair result after a game in which both sides showed plenty of attacking intent but little composure in front of goal.
Benteke, on as a half-time substitute for Divock Origi, broke the deadlock with a powerful header from James Milner’s cross.
But Southampton drew level when Gaston Ramirez headed across goal and Mane got the final touch. Mane was sent off in stoppage time for a second bookable offence but there was no time for Liverpool to capitalise on their numerical advantage.