Brian O'Neil was the unlucky player to have his penalty saved by Barnsley substitute keeper Nick Colgan.
Preston had most of the play in the first half with David Nugent having a good shot saved after just five minutes, but Barnsley's Chris Shuker also forced a save out of Carlo Nash after Matt Hill had given the ball away.
Patrick Agyemang hit the crossbar after his shot was deflected over the keeper after a David Jones through ball and Lewis Neal had a shot straight at Scott Flinders.
Just before half0time Barry Conoln had a diving header saved by Nash.
Nugent hit the left-hand post in the 51st minute.
Substitute Danny Dichio made it 1-0 to Preston after 77 minutes. From Paul McKenna's pass he hit a 12-yard right-foot shot into the bottom left.
Barnsley substitute Burns equalised in the 81st minute, a five-yard left-foot shot into the bottom right of the net after a cross by Robbie Williams.
The equaliser took the game to extra-time and in the 96th minute Dichio had a shot saved by Flinder,s who tipped the ball around the left-hand post for a right-wing corner but twisted his back and had to be substituted by Colgan.
Burns made it 2-1 to Barnsley after 107 minutes. A four-yard right-foot tap-in after a left-wing cross by Paul Hayes.
The referee awarded Preston a penalty for a handball by Matt Carbon and two minutes from the end of extra-time Alexander sent the keeper the wrong way with a right-foot penalty into the bottom right to make it 2-2 and send the game to penalties.