The prolific Charlie Austin had earlier got on the end of Danny Lafferty's cross to net for the 26th time this season as the home side went into a ninth-minute lead.
After the Clarets had a strong penalty claim turned down deep into a fragmented second half, Polish midfielder Cywka produced a moment of quality to level late on.
Cywka started from the bench alongside Jim O'Brien and Ryan Tunnicliffe as former Burnley defender Stephen Foster, Jacob Mellis and Marlon Harewood all earned a starting place for the visitors.
Sean Dyche made no changes to his side following Saturday's 1-0 victory at Charlton.
Austin's sensational effort secured the points in that game at The Valley, and the Clarets' top scorer was on the mark again early on.
After good work from Lafferty, Austin met the Northern Irishman's first-time cross with a deft clipped finish from close range to give the hosts the advantage.
That was the Lancashire club's first home goal since January 26 but it was the away side who looked most likely to make the next breakthrough.
Only a last-ditch block from Jason Shackell prevented Harewood from levelling the scores 10 minutes later after Dyche's men failed to clear Tom Kennedy's corner.
The Tykes continued to press for an equaliser with Chris O'Grady, who scored twice for Sheffield Wednesday at Turf Moor in October, flashing his 25th-minute effort from the edge of the area just wide of the left-hand post.
Barnsley's best chance of the first half fell to Dickson Etuhu who found half a yard of space inside the area before directing a powerful header on target which brought a great parried save from Lee Grant.
David Flitcroft made a half-time switch with Cywka replacing Harewood and the new 4-4-2 formation almost paid immediate dividends with O'Grady flicking on to Chris Dagnall who should have done better, firing wide with a clear sight of goal.
Flitcroft's opposite number then made a double change as the Clarets attempted to secure their slender advantage.
The home side looked to have earned a penalty with just 15 minutes remaining but referee Carl Boyeson waved away Austin's protests after he had been brought down by Martin Cranie.
Cranie and his team-mates were given another let off seconds later as Austin missed the target when well placed with a left-footed strike.
With six minutes remaining, the Turf Moor crowd's sense of injustice was compounded as Cywka found the top corner with a sweet strike beyond Grant as the visitors avoided three consecutive defeats.