The Black Cats played with 10 men for 70 minutes after defender George Brady was shown a red card in the opening stages of the match.
Burnley took the lead at Eppleton just before half-time through Jimmy Dunne after Sunderland failed to clear a corner.
Spirited Sunderland equalised eight minutes from time through George Honeyman, who was on the first team bench against Watford on Saturday.
Ethan Robson, also an unused sub against the Hornets, started the game but there was no place for 17-year-old striker Joel Asoro.
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Asoro, Elliot Embleton and Josh Maja were left out following their involvement in the FA Youth Cup victory over Burton Albion on Thursday evening.
Sunderland were controversially reduced to 10 men with just 20 minutes played.
Burnley's Khius Metz broke free and Brady tugged on his shirt outside the area, referee Steven Rushton waved play on, Metz's shot saved by James Talbot.
Rushton then awarded the free-kick for the pull and sent Brady off for denying a goalscoring opportunity, despite the striker having an attempt on goal.
Sunderland faced an uphill challenge after Dunne's opener but battled back well and equalised through Honeyman, who broke into the box and found the bottom corner.
Sunderland had further chances to win the game but in the end both sides settled for a point.
After three games - the halfway point of the group stage - Sunderland Under-23s are a point clear of second-place Blackburn Rovers in Group B after a win and two draws.
Sunderland U23s: Talbot, J Robson, Brady, Ledger, T Robson, Beadling (Casey, 58), E Robson, Lawson (Pybus, 82), Nelson, Molyneux (Blinco, 84), Honeyman.
Subs not used: Pain, Wright
Booked: T Robson
Sent off: Brady
Burnley U23s: Mitchell, Clarke, Olomewewe, Dunne, Birch, Bayode, Howarth (Koiki, 87), Jackson, O’Neil, Chakwana (Jumeau, 70), Metz (McNeil, 29).
Subs not used: King, Yao
Referee: Steven Rushton