The former St Johnstone midfielder kept a clear head to net a fine treble while his team-mates did their best to let the League One upstarts back into the game after they had eased into a 4-1 lead.
Hibs entered the competition a round later than the majority of their Scottish Premiership counterparts on account of their participation in what turned out to be a wretched European campaign.
Thankfully for manager Pat Fenlon, five games without defeat going into the third-round meeting with the Blues eased the pressure on his position and helped fade memories of that infamous 7-0 Europa League drubbing by Malmo.
But Stephen Aitken's Blues had pulled off the shock of the previous round by knocking out top-flight Ross County and they looked set to repeat the act in Edinburgh as Jamie Longworth's header gave them a fourth-minute lead.
Hibs did not panic at that point and were level 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 Fenlon's side were soon teetering again.
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 yet another 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.
Saturday's 2-0 win over St Mirren and the vivacious display which took care of the Buddies went some way to repairing the early damage done to relations with the Easter Road support following Hibs' feeble efforts earlier in the season.
Yet the fans' patience looked as if it may be tested once more as the visitors snatched a shock lead.
Putting a man on the post at set-pieces is just common sense but the home side left the back post uncovered and Longworth took full advantage as he directed Aitken's near-stick corner beyond the stationary Ben Williams.
Had this game come back in early August, you would have fancied a full-scale collapse for Fenlon's team. But their recent performances has refilled their reserves of confidence and the fourth-minute set-back was not allowed to derail them for long as Rowan Vine found Craig darting into the box and the midfielder sent a low header beyond Stranraer keeper David Mitchell's desperate dive.
Mitchell did better on 18 minutes as he pushed away Michael Nelson's downward header from Craig's deep free-kick while he was down well to block from Zoubir following the French winger's mazy run.
Zoubir had waited almost a month to make his first start for Hibs after a deadline-day move from French Ligue 2 club FC Istres but his fancy footwork was quickly winning over the home support.
And he confirmed his new-found crowd favourite status after 33 minutes as he cut in from the left before sending a right-footed strike in off the far post with a helpful touch from Blues midfielder Grant Gallagher's head along the way.
Hibs were knocking the ball about with the same verve they showed in beating Danny Lennon's side at the weekend and it was another tempting ball from Vine which fooled Rumsby into throwing a leg out in his own six-yard box, sending the ball past the stranded Mitchell.
Craig's second - struck when he latched onto another excellent delivery from Vine before poking into the corner of the net two minutes after the break - should have been that for the hosts.
But Aitken was allowed to sweep home a fine finish 60 seconds later before a crazy seven-minute opening to the second 45 was completed by Nelson inexplicably heading into his own net from the visitors' corner delivery.
Thankfully for Fenlon, sense soon prevailed as Rumsby handled a James Collins shot while grounded in the box, allowing Craig to smash the spot-kick straight up the middle to wrap up his first career treble.
But credit it to the plucky lower-league Blues who resisted the urge to accept defeat and caused a few more moments of concern for Hibs as Aitken twice came close to beating Williams.
- Sports & Recreation
- Pat Fenlon
- Stephen Aitken