Scott Allan, in his third stint at the club, put the Easter Road side ahead from the penalty spot in the final minute of the first half before David Wilson equalised for Stirling midway through the second period.
That took the tie to penalties and Hibs eventually came out on top 5-4, with Stirling substitute Craig Truesdale missing the decisive kick in sudden death.
All six of Hibernian's summer signings had started the game, while their new American owner Ron Gordon was an interested spectator as he took in Hibs' first competitive match since his takeover a few weeks previously.
All eyes were on Christian Doidge after Hibs paid £300,000 to sign the striker from Forest Green Rovers and the Welshman almost made a dream start at this new club after 20 minutes. Daryl Horgan picked him out with a terrific cross and, although Doidge's header was on target, goalkeeper Blair Currie made a terrific stop to keep it out.
Hibs should have moved in front seven minutes before the interval. Horgan sent over another tempting cross but Joe Newell, another debutant, somehow squirted his header off target from only a handful of yards.
They did get their goal a minute before half-time and in contentious circumstances. Stevie Mallan's free-kick on to the arm of defender Paul McLean was deemed a penalty under new regulations and Allan stroked home to kick under Stirling protests.
Hibs passed up several good chances to extend their lead and paid the price when Stirling drew level after 69 minutes, Wilson curling a shot into the top corner from 25 yards.
Neither side could find a winner sending the tie to penalties.