Burnley settled the quicker of the two sides, despite the absence of the suspended Jon Harley and James O'Connor, and created the first opportunity as Micah Hyde crossed from the left for striker Gifton Noel-Williams to head well wide.
Burnley went in front as John McGreal won possession on the left touchline and passed to Steve Jones who picked out Wade Elliott who then found Kyle Lafferty who confidently beat keeper Darren Ward.
Jones then created an opportunity for himself but spoilt his approach play by blazing over from 18 yards.
Sunderland's first shot on target came from Stephen Elliott who tested Danny Coyne on his near post, while at the other end Gifton Noel-Williams headed wide at the far post after a cross from Wayne Thomas.
Sunderland almost equalied from a Ross Wallace corner as Stephen Caldwell 's header was cleared off the line by Lafferty.
Lafferty could have scored again prior to the break after latching on to a throughball from Hyde but curled his shot wide.
The Clarets started the second period strongly as Jones got to the by-line and crossed to the far post where Wade Elliott volleyed inches wide.
The home side didn't have to wait too long to increase their lead as Lafferty seized on to a mistake by Dwight Yorke and buried his shot past Ward.
Wallace drove over from 25 yards before being replaced by Liam Miller while Yorke made way for Graham Kavanagh.
Lafferty maintained the two-goal advantage by again clearing off the line, this time to deny Stanislav Varga.
Sunderland's fightback began when Grant Leadbitter fired home a beauty past Coyne from 20 yards.
Keane's side demonstrated their 'never-say-die' attitude by equalising in stoppage time with veteran striker Connolly showing his class to score the equaliser from fully 25 yards.
The draw means that Burnley have still not beaten Sunderland at Turf Moor since 1973.