If Bradford were affected by the penalty miss it did not show as they scored the opener in the 10th minute.
Bradford's lead did not last long, though, as Jonathan Obika headed home from close range after Bradley Smith's corner was met by Michael Smith, who nodded across for Obika to convert in the 23rd minute.
Obika doubled his tally and Swindon's lead in the 43rd minute when Bradford failed to clear a corner, letting the ball drop to the former Tottenham Hotspur striker, who smashed home on the stroke of half time.
Shortly after the break, Obika was denied his hat-trick when his effort was blocked by the recovering Sheehan.
Left-back Sheehan seemed to be at the heart of all Bradford's attacks and he almost provided an equaliser for James Hanson, but the tall striker glanced his header wide.
Sheehan was the provider once again in the closing moments, crossing for Stephen Darby to force Foderingham into a fine save.
The Irishman was at the centre of things again in the dying moments, but this time for the wrong reasons as he was sent off for what appeared to be a punch on Raphael Rossi Branco.