Birchall came on in the 68th minute with his side a goal down through Blair Sturrock's opener on the hour.
Birchall latched onto a clearance by Barnet goalkeeper Rob Beckwith and fired the ball into the top left-hand corner from 15 yards to earn his side another chance to get through to the FA Cup fourth round and meet one of the Premier League giants.
Swindon started the game brightly and should have opened the scoring in the fifth minute through mis-firing striker Billy Paynter.
Paynter latched onto a cross shot from Sofiane Zaaboub but could only head straight at Beckwith with the goal at his mercy.
Paynter has not scored for 11 games since he netted in the first round of the cup against Wycombe back in November.
He turned provider in the 28th minute when he played in striker partner Sturrock, who drew a good save from Beckwith with a shot from ten yards.
In the 40th minute, Jamie Vincent crossed to Paynter who again missed the target with his header from 12 yards out.
Swindon took the lead in the 60th minute through Sturrock's fourth goal of the campaign.
Right-back Miguel Comminges cut into the penalty area before firing in a cross shot that Beckwith could only palm up into the air.
After a goalmouth scramble, Sturrock bundled the ball home from two yards out.
Birchall came on for Ashley Carew in the 68th minute as Barnet looked for an equaliser and they got what they wished for in the 85th minute.
Goalkeeper Beckwith hoofed the ball down field and the ball fell to Birchall inside the penalty area.
He turned an fired the ball home into the top left-hand corner from 15 yards for his fifth goal of the campaign.
Swindon almost stole the victory through substitute Kaid Mohamed when he slalomed through the Barnet defence before firing his shot just over the crossbar.