Southend were level on aggregate shortly before the break after another failed clearance allowed Anthony Straker to pounce from close range.
Leonard had a chance to add his second of the afternoon shortly after the restart when he tried to drill the ball in from the edge of the area, but Lyness was able to deny him.
Keen to retake the lead, Burton brought on Billy Kee, who looked dangerous straight away as he tested Bentley with a deflected shot from just inside the box.
The visitors continued to look the more likely of the two sides and were able to restore their lead when Southend struggled to clear a free kick, allowing the ball to drop for McGurk, who had been left unmarked, and was able to fire in low for his second goal of the tie.
Alexander MacDonald had a chance to extend Burton's advantage further shortly after, but Bentley managed to deny the striker.
Desperately seeking a late equaliser, the Southend keeper went up for a corner with over four minutes still to play, but the extra man did not give the home side enough to find their third goal of the afternoon.
Leonard came close in stoppage time as he fired the ball towards goal, but his finish drifted a couple of inches wide, while a powerful effort form Jack Payne was saved by Lyness.
Burton will face Fleetwood Town in the final later this month after they beat York City.