His 88th-minute winner spared Swindon's blushes after Blue Square Premier League side Forest Green Rovers had threatened an FA Cup shock.
Sturrock's winner also silenced Swindon's angry fans, who protested long and loud after the match demanding changes in the boardroom at the troubled County Ground club.
A 3-2 defeat was rough justice on Forest Green, who led 2-1 midway through the second half.
They initially trailed to a 12th-minute inswinging corner by Jon-Paul McGovern and were fortunate as both Christian Roberts and Barry Corr missed chances for Swindon before the break.
But Forest Green, who knocked Rotherham United out in the last round, got a deserved equaliser six minutes into the second half when top-scorer Stuart Fleetwood buried his 24th goal of the season.
It was all hands to the pump as Swindon clung on but they then fell behind when Mark Beesley headed Fleetwood's 65th-minute corner into the roof of the net.
Forest Green's joy was short-lived though as the influential Hasney Aljofree stole in at the near post to head Swindon level.
Sturrock came off the bench seven minutes from the end to snatch Swindon's fortuitous winner - an initial effort by Billy Paynter having been blocked.