Matt Ritchie's long-range speculative effort with 14 minutes left of the third-round tie struck the legs of Benson, a January signing from Charlton, and rolled into the back of the net to hand the Robins a famous victory.
Earlier, the visiting Latics had taken a 34th-minute lead after Callum McManaman followed up to beat Wes Foderingham in the Swindon goal after Ben Watson's penalty had struck a post. But the hosts showed great self-belief to pull themselves back into the game through Alan Connell's headed equaliser before Benson won it late on.
Saturday's match marked the first time the Robins had entertained Premier League opposition in the FA Cup since Coventry came to Wiltshire in January 2001.
That was the same year current Swindon manager Paolo Di Canio etched his name into the competition's folklore, slotting past Manchester United goalkeeper Fabien Barthez and the Frenchman's infamous hand in the sky to win a fourth-round tie for West Ham at Old Trafford.
And the Italian was hoping to conjure up a little more magic against their out-of-form and out-of-luck opponents.
In the 12th minute Di Canio's men should have taken the lead as Raffaele De Vita seized on an errant Adrian Lopez pass and picked out Ronan Murray, only for the on-loan Ipswich striker to fire at Ali Al Habsi.
Hendry Thomas blazed over the Swindon bar at the other end a minute later, while Paul Caddis' deflected cross almost embarrassed Al Habsi - who was grateful to see it skid past his right-hand post.
Full-back Callum Kennedy tried his luck from distance for the Robins without reward after 21 minutes and Franco Di Santo did likewise for the visiting Latics moments later.
Although Wigan continued to dominate possession, Swindon created their fair share of chances and Jonathan Smith dragged his effort from 20 yards well wide of the right-hand post.
However, the visitors broke the deadlock after 34 minutes.
Aden Flint felled McManaman in the area, referee Peter Walton pointed to the spot and, after Watson smashed the resulting spot-kick against a post, McManaman was on hand to slip the ball beneath Foderingham.
Swindon responded well to falling behind and only the outstretched hand of Al Habsi denied winger Ritchie an equaliser three minutes after the hosts went behind.
And with five minutes left until half-time, the Robins got themselves back on level terms.
Ritchie's pin-point cross from the right picked out Connell, whose glancing header nestled in the bottom right-hand corner.
Wigan fans thought they regained the lead two minutes later when Shaun Maloney had the ball in the back of the net, but the effort was disallowed for offside, while Murray sent a 25-yard effort narrowly over just before the break.
In front of a sell-out crowd who had come to the County Ground in hope of an upset, Swindon settled into a good rhythm after the interval.
Di Canio swapped Murray for Benson at half-time and he quickly formed a decent partnership with Connell, who poked Kennedy's cross wide of the near post after 50 minutes.
Connell wanted a penalty when he went down under pressure from Gary Caldwell, but referee Walton remained unmoved by the striker's protests.
On the hour mark Ritchie missed a glorious chance to hand the underdogs a shock lead.
A neat passing move involving Connell, Smith and Benson ended with the latter picking out the unmarked Ritchie eight yards from goal.
However, the former Portsmouth man could only flash his effort high and wide when well placed.
Wigan could not muster a shot on target in the first 15 minutes of the second period, and when they eventually did McManaman's deflected effort was comfortable for Foderingham to parry round a post.
Swindon were evidently starting to run out of energy, and a series of Wigan attacks seemed to be draining the Robins of their remaining resources.
But Lady Luck shined on the hosts with 15 minutes remaining and from somewhere they found a second goal.
Ritchie collected Simon Ferry's pass 25 yards out and let fly with a low drive which struck the legs of Benson and rolled into the net with Al Habsi prostrate on the floor.
Ferry sent an effort from the edge of the area high and wide and Ritchie nodded substitute Alessandro Cibocchi's chipped centre well over, as the Robins' defence stood firm at the other end to seal another memorable FA Cup moment for Di Canio.