Mark Birch went with an unchanged team from the first round victory over Altrincham a couple of weeks ago.
Barnsley created the first chance of the game with ten minutes on the clock when Randall came out to clear but didn't get the connection he wanted. Afuape thought he had an open goal to aim at but Hayden Atkinson got back to make an fantastic block.
The United 'keeper was called into action again shortly afterwards when Barnsley broke down the right. The cross was pulled back to Alker on the edge of the box but it was pushed round the post well.
Barnsley kept the pressure on and created another chance after a corner was only half cleared. Ogbu got a cross in from the left and Afuape got on the end of it, but Thomas Randall in United's goal was equal to it again.
United worked their way into the game and earned their first corner on the half-hour mark. Lowes and Hopper worked it short but Skinner's pass into the box was cu out and Barnsley got it away.
Kieran Graham should have put the hosts ahead just before the break when he connected with a cross from the left on the penalty spot, but he fired the shot high over the bar to send the sides into the interval level.
HALF-TIME - BARNSLEY 0-0 UNITED
The Reds had a great chance in the opening thirty seconds of the second half when another cross came in from Ogbu on the left, it was awkward for Afuape to control and the shot landed on the roof of United's net.
United took the lead with 51 minutes on the clock when half-time substitute Lewis Lambert's shot slipped through the Barnsley keeper and went into the back of the net after Penney won it back inside the opposition half.
Barnsley almost replied instantly when Kieran Graham took the shot on from 20 yards, but it cannoned off Thomas Randall's post and United got it clear.
Randall pulled off a fantastic double save a minute later when he got a big hand to the initial shot from Alker before throwing himself at the feet of Afuape to prevent the rebound going in.
A dangerous Chapman cross was then sliced behind for a United corner at the other end. A training ground routine saw Lowes play it low to Rigby on the edge of the box but his shot was blocked by a defender.
The final action of the game was a Barnsley corner which Randall came out to punch superbly again to send United into the third round.
FULL-TIME - BARNSLEY 0-1 UNITED
Teams:
Barnsley - Wilson, Tommy-Mbogba, Ogbu (J Woodcock 62)', Kay, Thompson, R Woodcock, Warburton, Probert (Reid 66)', Afuape, Graham, Alker.
Unused subs - Andrassy, Price, Rayner, Daniels, Cutler.
United - Randall, Chapman, Hodgson (Smith 78'), Penney, Porter, Atkinson, Rigby, Skinner (C) (Lambert 46'), O'Donoghue, Lowes, Hopper.
Unused subs - Sillitoe, Bell, Hatton.
Goals:
Barnsley - N/A
United - Lambert (51')
