The Tykes took the lead early on in Tuesday night's Championship clash at Elland Road when Jason Shackell powered home but the hosts started to show their class as the half wore on and they equalised through Luciano Becchio.
Max Gradel was awarded, and then stepped up to convert, a controversial penalty in the 37th minute before Matt Hill dragged the visitors level.
Bradley Johnson was then dismissed for a second bookable defence and it looked like the game was slipping away from Leeds.
However, Gradel capitalised on Shackell's error to put Simon Grayson's men in front once again.
Dogged Barnsley refused to give up though and Trippier's 25-yard free-kick ensured the spoils were shared.
The Tykes were quickest out of the traps as they broke the deadlock inside the first two minutes.
Trippier's right-wing cross was cut back into the centre by Danny Haynes at the back post and Shackell followed up to blast past a helpless Kasper Schmeichel.
The visitors were on the receiving end of some luck moments later when Gradel's 25-yard free-kick came back off the crossbar and hit goalkeeper Luke Steele on the back of the head before going inches wide.
However, Leeds are the highest scorers in the league and their persistence paid off as they drew level in the 23rd minute - thanks, in part, to an error from Steele.
The Tykes shotstopper failed to get anything on Neil Kilkenny's inswinging left-wing cross and Becchio pounced at the far post to head into an empty net.
The hosts were given a major let-off when Garry O'Connor volleyed over from a good position and they capitalised on their luck even further when they were awarded a fortuitous penalty.
Gradel, who was causing no end of trouble for the Barnsley defence, burst through on goal and an innocuous looking challenge from Hill was deemed to be an unfair one by referee Mark Clattenburg, who promptly pointed to the spot.
Gradel got up to take the kick and he made no mistake by sending Steele the wrong way to put his side in front eight minutes before the interval.
Hill atoned immediately after the break though as the visitors, once again, came out all guns blazing.
The defender powered forward from defence and latched on to Jacob Mellis' through ball before calmly slotting past Schmeichel in the 48th minute.
Leeds' task was made even harder when they were reduced to 10 men two minutes later as Johnson was dismissed for a second bookable offence following an ill-timed challenge on O'Connor.
Schmeichel saved well at Haynes' feet as it looked Barnsley would seize the initiative.
However, after dispossessing Shackell on the edge of Barnsley's area, Gradel grabbed his second with a lovely curling effort that found the far corner of the net in the 70th minute.
To their credit, Mark Robins' men refused to give up and they hauled themselves level once more eight minutes from time when Trippier's tame free-kick found its way through the Leeds wall and past an unsighted Schmeichel.