Dagnall put in a man-of-the-match performance on a crisp evening in North Lincolnshire to continue Wednesday's winless run in Scunthorpe that stretches back to 1958.
Michael Collins' curling effort and Michael O'Connor's penalty, coupled with Dagnall's goals, secured the Iron's passage into the third round for the second successive season.
Wednesday played the majority of the second half with 10 men after former Iron Richard Hinds was given his marching orders, with Marcus Tudgay and Neil Mellor scoring the visitors' consolation goals.
Scunthorpe headed into the clash nursing a couple of knocks meaning Collins and skipper Cliff Byrne came into the starting line-up, while the visitors made six changes to the side that defeated Brighton 1-0 at the weekend.
The Iron came out on top when the duo locked horns in a pre-season friendly under a month ago, but neither showed the skill or guile required to make a breakthrough in the opening exchanges.
The first half was a drab affair but burst into life on the stroke of half-time when Collins scored a splendid curling effort from 18 yards.
Scunthorpe continued in the same vein after the break and extended their advantage four minutes into the second half when Dagnall wriggled free just outside the area and unleashed a superb drive into the bottom corner.
The Iron pressed forward and soon had the chance to make it three when Hinds clattered Jonathan Forte in the six-yard box.
Referee Nigel Miller had no hesitation in awarding a penalty and sending off the ex-Iron defender, with O'Connor scoring the resulting spot-kick.
Tudgay gave the visitors a lifeline from the penalty spot, before Dagnall headed home Byrne's superb right-wing cross with eight minutes remaining.
Substitute Neil Mellor reduced the deficit a minute later but the Iron held strong to send home the 1,860 visiting fans disappointed.