Jamie Longworth's header gave Stranraer a fourth-minute lead but Hibs levelled within moments as Craig nodded home his third of the campaign before Abdellah Zoubir's deflected strike and a Scott Rumsby own goal had them clear at the break.
Craig added a fourth within two minutes of the restart but Chris Aitken's 20-yard curler and a quick-fire third for Stranraer - another OG, this time by Hibs defender Michael Nelson - was enough to have the Edinburgh men fearing a mortifying defeat.
But Craig's third from the penalty spot on 61 minutes calmed fraying nerves and Hibs held out to secure a place in the last eight.
Stevie May returned to haunt former club Hamilton as he bagged a double to send St Johnstone into the next round after a 3-0 triumph.
May, who spent last season on loan with Accies, grabbed the opener early on in the tie at New Douglas Park.
Gwion Edwards then came off the bench to claim his first goal for the club late on, before May struck again in injury-time.
Billy McKay netted his seventh goal of the season as Inverness beat Dundee 1-0.
McKay had already threatened with a shot over the bar before giving Caley Thistle a ninth-minute lead at Dens Park as they bounced back from their first defeat of the season against Aberdeen.
The visitors worked the ball well down the left side and Richie Foran gave himself a shooting chance following a one-two with Nick Ross.
The captain sliced his shot but the ball fell kindly for McKay eight yards out and the Northern Ireland international reacted quickly to knock the ball home.