First-half goals from Dagnall and 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.
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.
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.
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.
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.
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.