Supporters queued up for hours for tickets and the club’s website experienced problems dealing with demand from fans hoping to see their team win the cup for the first time since 1957.
A further batch of 2,000 tickets will be available from 10 o’clock this morning.
Meanwhile, Rory Loy is on course to be fit for the final – despite suggestions he may have suffered a broken leg.
The striker has been sidelined since damaging his calf in Falkirk’s quarter-final triumph against Queen of the South on 6 March, but has been through an intensive recovery programme in recent days, working one-on-one last month with the physiotherapy staff at the FA’s national football centre at St George’s.
Loy is to return to the state-of-the-art Staffordshire facility this week to step up his rehab and Bairns boss Peter Houston is hopeful.
Houston said: “They think there might well have been a slight fracture, because there’s calcification around the bone area, but the head physio thinks there’s a huge chance he could be ready for the final, which is a massive boost for us.”