Rangers, who have just won promotion back to the Scottish Premiership after four years exile in the lower divisions following their financial troubles, clinched the tie when Tom Rogic blazed his penalty high and wide to send the champions-elect out.
Rangers, who won the Championship (second tier) title and promotion this month, will face Hibernian in an all-Championship final at Hampden on May 21.
Rangers went ahead after 16 minutes when veteran striker Kenny Miller, 36, one of only five players since World War Two to play for both clubs, took advantage of poor Celtic defending to turn and shoot past goalkeeper Craig Gordon.
News that makes sense
Your trusted source for staying up-to-date with the world around you. Get free daily news updates and analysis, straight to your inbox.
Celtic, who had been second best for most of the first half, took control after the restart and substitute Erik Sviatchenko equalised with a bullet header after 50 minutes.
Barry McKay swung the match Rangers way with a stunning 30-metre strike in the sixth minute of extra time, before Rogic scored Celtic's second equaliser 10 minutes later.
Hibs secured their place in the final when they beat Dundee United, the bottom club in the Premiership, 4-2 on penalties after the their semi ended 0-0 after extra time at Hampden Park on Saturday.
(Reporting by Mike Collett, editing by Ed Osmond)
