That will be Wigan's first-ever appearance in a major final. The Premiership's surprise of the season did win the Freight Rover Trophy in 1985, but it hardly compares to tonight's triumph.
Roberts, tall and powerful, is the sort of striker that Arsenal would never sign. But he terrified their defenders and finally finished the last of a series of chances.
This was a triumph for Wigan's strength and tenacity after Robin van Persie's 108th-minute free kick seemed to have earned Arsenal their first final place since 1993.
The Gunners have only themselves to blame. Jose Antonio Reyes missed an early penalty, Arsenal's fifth penalty miss of the season and their dilettante approach was in stark contrast to Wigan's hard work. Arsene Wenger has yet to learn that there are other values in football besides skill and pace.
Arsenal have rarely taken the League Cup seriously during Wenger's nine and a half years in charge, but their season so far has been the worst of his reign. They are in danger of missing Champions League qualification and the League Cup represented an insurance policy into Europe.
That is why Wenger fielded Sol Campbell, Dennis Bergkamp and Thierry Henry as his side attempted to pull back Wigan's 1-0 first leg lead. It was Henry's first League Cup appearance for more than six years.
Wigan were without the injured Arjan de Zeeuw, Emerson Thome, Ryan Taylor, David Connolly and Lee McCulloch who could postpone his hernia operation no longer. But they did field on-loan Neill Mellor, with Reto Ziegler among their subs.
Wigan made the sharper start. Their tactics were obvious: to rattle Arsenal physically. The Gunners, and their manager, hate that approach.
Reyes set up Arsenal's first chance, Henry's weak seventh-minute effort wide.
Roberts turned Philippe Senderos and fed Alan Mahon who scuffed his shot.
After 14 minutes, Henry, breaking on the left, found Aleksandr Hleb, but Stephane Henchoz deflected his shot wide.
In the 21st-minte Henry crossed from left, Henchoz handled and Mike Pollitt pushed away Reyes' penalty. It was the first of a series of brilliant saves.
Pollitt beat away Bergkamp's crisp shot a minute later, then palmed out Henry's curling shot. Wenger's frustration began to show on the touchline.
Arsenal were lucky to escape a penalty when Senderos hauled back Roberts.
Pollitt easily held a weak Henry free-kick, then saved low from the Frenchman as he burst on to Bergkamp's incisive pass. But they were all from outside the box.
Almunia saved at Roberts' feet while Wigan claimed another penalty. Gary Teale's follow-up was hacked away. Roberts forced a great save from Almunia in first-half stoppage time.
Pollitt turned away another Henry curler two minutes after the restart. But Arsenal's intricate attacks lacked variety. As ever, their goals have to be threaded through the eye of a needle. They are incapable of playing a simple cross to runners in the box.
Wigan do. Roberts flashed a header over from Teale's cross.
But Henry headed the leveller in the 65th minute. Hleb's pass split the defence. Kerrea Gilbert raced on and hit a pinpoint cross for Henry to head home.
Almunia had to save low from Andreas Johansson as Wigan pressed again in the air. Wigan appealed for another penalty as Pascal Chimbonda went down under Almunia's challenge.
Senderos flashed a header wide as the Gunners pressed for a winner. Robert Pires' shot was deflected to Pollitt.
Wigan kept coming. Roberts hit wide again after Senderos fell over as Teale crossed. Henry limped off with a hamstring problem.
Bergkamp shot into the side netting under Graham Kavanagh's challenge. Campbell blasted an 86th-minute chance over after a half-cleared corner fell to him.
Campbell made a last ditch extra time tackle on Roberts after another mistake by Senderos. Pollitt pulled off another vital save as Bergkamp smacked Reyes' through ball.
Then Van Persie struck and Arsenal thought they were through. Pollitt denied Bergkamp a third goal. Then Roberts struck the final blow.