Billy McKay looked to have calmed any nerves in the visitors' ranks with the opener midway through the first-half but Brown Ferguson's penalty just before the break ensured the tension was ratcheted up as time slipped by.
The inevitable penalties duly arrived but, with Richie Foran, the Inverness captain, and Stenhousemuir midfielder Bryan Hodge both missing, the spot-kicks went to sudden death and visiting goalkeeper Ryan Esson proved the hero with his save from Darren Smith's effort.
Stenhousemuir did not have to look far for inspiration, with their passage to this stage of the competition being earned via a shock defeat of holders Kilmarnock in the previous round. As the rain and wind swept across the artificial surface at Ochilview, the hosts were clearly optimistic of taking another scalp – but fell just short.
Robbie Thomson, the young Celtic goalkeeper, answered an SOS call earlier in the day to join the second division outfit on loan and the teenager could be happy with his efforts on a night when his new team played some superb football.
Thomson dealt with his first test soundly in the 13th minute when he dropped confidently on to Aaron Doran's low shot. However, there was little the 19-year-old could do in the 23rd minute as Inverness broke the deadlock – a deflection on Foran's threaded through ball caught the home defence flat-footed and McKay pounced to nod over Thomson from 16 yards out.
Terry Butcher's side appeared to have proceedings under control at this point but let their guard slip six minutes before the interval. David Raven was guilty of dallying as Stenhousemuir burrowed their way into the box and, as the full-back finally swung his boot at the ball, he only succeeded in catching the leg of Ferguson. The midfielder, who scored from the spot in the win over Kilmarnock in the last round, duly found the back of the net again.
The part-timers then came within a whisker of taking a half-time lead. Sean Dickson's right-wing cross was volleyed by Stewart Kean and, with the effort holding up in the wind, Esson watched it bounce off the top of his crossbar.
Buffeted by playing into the stiff breeze, Inverness enjoyed the lion's share of possession after the interval but Andrew Shinnie's deflected shot was as close as they came going into the final 20 minutes. With 15 minutes left, Foran thought he had won it, but his powerful header was superbly touched over by Thomson.
Stewart Kean almost won it for Stenhousemuir in injury time, but his shot flashed past the post. Extra-time duly came and went with the drama of the shoot-out finally going the way of the top-flight side.