Marlon Harewood was also on the scoresheet for David Flitcroft's side, who survived a second-half comeback to book its place in the last eight of the FA Cup.
The visitors roared out of the blocks, and, inspired by midfield playmaker Jacob Mellis, established a two-goal lead through Dagnall and Marlon Harewood inside 20 minutes. Lifeless before halftime, the Dons improved after the break, but Dean Bowditch's goal was all they could muster, and Dagnall added a late third on the counterattack to seal the win.
Mellis operated behind strikers for the Tykes and proved an elusive customer. As early as the fourth minute, he released Dagnall with a chipped pass that dropped behind Gary MacKenzie, and the striker finished coolly across goal with his left foot.
Dagnall peeled off his marker for a similar chance before the 15-minute mark but volleyed narrowly over the crossbar from a difficult angle. The Dons' respite was brief, though, as Mellis instigated another slick move that led to the second goal of the game.
The former Chelsea youth player picked out the run of Scott Wiseman with a through ball into the penalty area, and the right back had plenty of time to assess his options before finding Harewood, who doubled the Tykes' lead from 12 yards.
Karl Robinson reshuffled his team into a 4-3-3 formation and while Darren Potter, sitting in front of the defence, helped to negate the influence of Mellis, the home team showed few signs of life in attack, with hopeful flick-ons by Alan Smith looking like their only route to goal.
Shaun Williams was moved from center back to central midfield and helped to improve the Dons in the opening stages of the second half before the game was halted for paramedics to treat Barnsley midfielder Kelvin Etuhu, who was stretched off after a nasty clash of heads with Adam Chicksen.
That did not slow their momentum, though, and soon after the delay Robinson's team halved the deficit. Bowditch stretched to knock a ball through the middle past center back Tom Kennedy and sped beyond the defender before slotting past Luke Steele.
Smith's aerial presence continued to increase in effectiveness, and Steele made a fine save to tip the striker's header over the bar. Soon after, he palmed Williams' long-range free kick around the post before Smith failed to keep his shot down from the subsequent corner.
Gaps began to open up at the back as the Dons poured forward, though, and Jason Scotland and Mellis might have done better with good chances to put the game to bed. Shortly after 12 minutes of added time had been indicated, though, Dagnall made no mistake, breaking the Dons' offside trap before rounding David Martin to ensure Barnsley's passage into the last eight before Smith saw a late red for an aerial challenge.