Brittain had put his side ahead from the spot five minutes before half-time, only for the effort to be cancelled out within three minutes by a headed goal from St Mirren striker Steven Thompson.
Brittain had a golden opportunity to score a stoppage-time winner after Mark Corcoran was brought down by Lee Mair, but this time he sent his penalty sailing well over the bar.
St Mirren, who now face their second replay in this season's tournament, welcomed back captain Jim Goodwin from injury for his first match since December 17.
His inclusion was one of three changes to the home line-up following the previous week's goalless draw with Inverness - Aaron Mooy and Gary Teale also came into the side in place of the injured Graham Carey, cup-tied Dougie Imrie and benched Steven Thomson.
Visiting manager Derek Adams handed starts to Steven Craig and Mark Corcoran, both of whom came off the bench in the preceding 3-0 win over Livingston, replacing Michael Gardyne and Rocco Quinn for the First Division leaders.
Colin McMenamin had the first half-chance of the game when he reacted well to a poor defensive ball in the box, but his shot after a turn was easily dealt with by home goalkeeper Craig Samson.
Paul McGowan was put through by a good ball but was robbed of possession as he tried to turn in the box, while at the other end Stuart Kettlewell tried a volley from the edge of the area but it was deflected off Goodwin and into the goalkeeper's arms.
St Mirren tried their luck at goal after 17 minutes when Jeroen Tesselaar sent in a ball from the left across to the opposite corner of the box, but it evaded David van Zanten who was aiming to connect at the byline.
Aaron Mooy stung the palms of County goalkeeper Michael Fraser with a speculative long-range effort from the left; and on the half-hour Teale weaved his way down the right flank but his cross was booted clear from the feet of David Barron.
County's Mark Corcoran sent an effort from just outside the box well over the bar after picking up possession in midfield - but his side then went in front from the penalty spot five minutes before the break.
Brittain picked out Stuart Kettlewell with a precise long crossfield pass and as the County midfielder controlled it and raced in at the right corner of the box, home goalkeeper Samson rushed out and brought him down as he attempted to sweep the ball away with his arm.
Brittain took the subsequent penalty and coolly fired it into the bottom left corner, with Samson diving the wrong way.
However, St Mirren were level within three minutes when Thompson rose to head home Van Zanten's cross from the right in a busy penalty area - and McGowan even went close to putting his side in front before the half-time whistle with a shot from outside the box which was palmed out for a corner.
St Mirren manager Danny Lennon began the second half with a double substitution - forward Nigel Hasselbaink and Thomson came on in place of Mooy and Teale respectively.
Both sides passed the ball well as they sought to take the lead. McGowan and Thompson linked well in a good move with the latter finally managing to find space to turn, but his eventual cross was easily claimed by Fraser.
Steven Craig just failed to connect with a ball into the box at the other end and Iain Vigurs then fired well over the bar.
At the other end, Hasselbaink fired in a promising cross from the left but Thompson was unable to direct his towering header.
Tesselaar produced a great challenge in the penalty box on the hour to prevent McMenamin shooting after good play involving Gary Miller.
Vigurs launched a half-volley from the edge of the area as the game entered its final quarter after a free kick had been cleared back out to him, but his powerful drive went wide as Samson scrambled to cover it, and McMenamin headed too high from Kettlewell's cross from the right.
Home scorer Thompson tried to double his tally 10 minutes from time with a volley from a cross by substitute Graham Carey - then McMenamin hit the side-netting after holding off Marc McAusland in a race into the box.
Carey stepped up to take a St Mirren free kick five minutes from time, but his curling 25-yard effort following Scott Boyd's push on Hasselbaink was punched away by Fraser.
Brittain had the chance to win the game three minutes into stoppage time after Lee Mair brought down Corcoran following a pacy counter-attack - but this time the County captain blasted his spot kick well over the bar with the quarter-finals in sight.