Sparkes won a replay for the Bees after his goal made it 3-3 on the night 15 minutes from time.
The 27-year-old beat off competition from the likes of Alexis Sanchez and Kevin De Bruyne to claim the accolade.
"I scored a similar one last year but this is a big stage, the fourth round of the FA Cup, full-house and on TV," he told the club's website.
"Me and JT [Jack Taylor] stepped over it and he said 'it's yours'. I got the ball and thankfully I hit it really well and it's gone in."
Barnet now face an away trip to Carshalton Atheltic in the FA Trophy on Saturday before their replay with Brentford next Tuesday.