With the game finishing in a penalty shootout, Swindon failed to convert any of their four penalties as Paul Fairclough's side set up a clash with Bristol Rovers on Saturday.
Billy Paynter's seventh goal of the season had given Swindon the advantage just before the break, but the same player put through his own net just after the interval as Barnet deservedly drew level.
And it was the home side that looked more likely to grab the winner until Michael Leary charged into a two-footed tackle which left Sofiane Zaaboub writhing in agony on the turf.
Beckwith made sure the home side forced extra-time after pulling off two point-blank saves from Paynter and Roberts, before performing heroics in the shoot-out.
After Adam Birchall's late equaliser at the County Ground, Barnet started with aspirations of making it through to the fourth round of the competition for the second year in succession.
With Josh Wright ineligible and Ashley Carew suspended, Paul Fairclough reverted to a 4-4-2 formation with Leary and Birchall given places in the starting line-up.
And with the confidence of an impressive away win at Milton Keynes still in their minds, the home side started brightly with Jason Puncheon testing Peter Brezovan with a rasping long-range drive.
It was Swindon's turn to look threatening just seconds later as Leary was forced into a last-ditch goal line clearance after Zaaboub's flashing effort looked to have beaten Beckwith.
Barnet continued to push forward and Cliff Akurang should have done better after taking Puncheon's pass in his stride, but the big striker failed to make the most of the opportunity as Swindon recovered to clear.
And the former Histon man was guilty of wasting another glorious opportunity - this time heading wastefully over the bar after Max Porter's delightful cross had found the striker unmarked just six yards from goal.
Barnet were punished for their profligacy in front of goal just before the break as Zaaboub's delicately chipped pass opened up the home backline, allowing Paynter time to pick his spot and fire low past Beckwith.
But Barnet didn't have to wait long to hit back as Puncheon's left-wing cross was inadvertently diverted into his own net by Paynter.
With the wind in their sails, Barnet continued to make a mockery of 29 league places between themselves and their League One opponents, with Puncheon at the heart of every move the home side created.
But with Barnet looking as if they would go on and win the tie, Leary decided to launch himself into a reckless two-footed challenge on Zaaboub - leaving the midfielder unable to continue while the referee Andy Penn needed little encouragement to show him the red card.
With just ten men, Barnet fought manfully to repel the Swindon attack, with Beckwith twice denying the visitors with impressive stops.
Both teams poured forward in extra-time looking for the winner, but with Barnet tiring, the home defence appeared to drop deeper and deeper.
But Barnet held out for penalties and while Swindon missed all four, the hosts made no such mistake and will now face Bristol Rovers in the next round.