First-half goals from Dagnall and Marlon Harewood ensured the tie went with the form book, with Dagnall adding a third late on as David Flitcroft's Tykes clinched their seventh win in their last eight matches.
The 2008 FA Cup semi-finalists survived a spirited second-half fightback from MK Dons, who pulled a goal back through Dean Bowditch but could not make further inroads.
The second period was marred by a serious injury to Barnsley's Kelvin Etuhu, before MK Dons striker Alan Smith was sent off in stoppage time.
The hosts had the first half-chance of the game when Luke Chadwick hit a speculative volley wide in the opening minute.
But it did not take long for Barnsley to take control as Dagnall - scorer of the only goal in the fourth-round win at Hull - latched onto Jacob Mellis' pass and fired an angled shot across David Martin with just 144 seconds on the clock.
The former Scunthorpe striker then had a good opportunity to double his tally but powered an effort over Martin's bar after easily springing the Dons' offside trap.
The npower Championship side were unbeaten in seven matches heading into the match and were soon comfortably in the ascendancy.
Harewood, an FA Cup runner-up in 2006 with West Ham, added the second goal after 19 minutes, side-footing home after being picked out inside the area by Scott Wiseman's clever cut-back.
Karl Robinson's side could not find a way into the game and were restricted to a few long-range shots, with Jon Otsemobor's wild volley 10 minutes before half-time epitomising the Dons' lack of attacking spark.
Jason Scotland was introduced by Flitcroft at the interval, with Harewood making way.
But it was the reinvigorated hosts who almost found a way through in the early stages of the second half as Dean Lewington's close-range header was deflected behind for a corner.
The game was halted for seven minutes shortly afterwards following a clash of heads between Etuhu and Adam Chicksen, which left the former requiring medical attention before being carried off on a stretcher.
The stoppage did not derail the Dons' lively start to the second half, and they halved the deficit through Bowditch just after the hour mark.
The 26-year-old striker kept his cool to slot past Luke Steele when he found himself one-on-one with the Barnsley goalkeeper to give Robinson's side renewed hope.
The visitors were still dangerous when they ventured forward, though, and Lewington was forced to hack a goal-bound effort off the line to keep his side in the tie with 20 minutes remaining.
Powell was causing problems from the right flank and again jinked inside a couple of challenges before seeing his cross cleared behind for a corner, but the set-piece was easily gathered by Steele.
The Tykes goalkeeper was then on hand to tip Smith's header over the bar as Robinson's team continued to ask questions of Barnsley's defence.
The set-piece bombardment of the visitors' penalty area continued and Steele was required to push a Williams free-kick around the post with the resulting corner bobbling around in the box before Smith, recalled in place of the cup-tied Izale McLeod in attack, lashed a volley over the bar.
Scotland then had a chance to make it four goals in four games for Barnsley since his move from Ipswich but the experienced forward thumped an effort high and wide into the large contingent of travelling fans behind Martin's goal.
Mellis lashed just wide moments later as Barnsley looked to seal the tie in the closing stages, with MK Dons' resurgence seemingly taking its toll on their players.
Referee Michael Jones added 12 minutes of stoppage time following Etuhu's injury, and Dagnall did not take long to put the result beyond doubt as he rounded Martin to finish off any hopes of a dramatic revival from the Dons.
Smith was then sent off for raising an arm against Rory Delap as Barnsley saw out the remainder of stoppage time to advance.