The hosts enjoyed the better of the chances in the opening exchanges as Jack Rodwell fired wide from a promising position in the early stages and then saw an effort fly narrowly wide from range, while Sam Vokes tested Vito Mannone in the Sunderland goal for the visitors.
Rodwell also saw a header well saved by Burnley stopper Nick Pope with 30 minutes gone in an entertaining first half.
Sunderland again started strongly after the restart but Burnley came into the game as the second half progressed but neither side could force a breakthrough.
The replay will take place at the Clarets’ Turf Moor home on Tuesday January 17.
Southampton have announced plans for a two-week tour of China as part of their pre-season preparations for the 2018-19 season.