Aaron Doran and Richie Foran had given Terry Butcher's men the goals their early enterprise deserved but Charlie Mulgrew pulled one back before the break before Matthews took centre stage late on as it finished 2-2.
Doran had already had one decent attempt on goal before he collected a ball 25 yards out and struck a superb strike past the helpless Fraser Forster (14).
The reigning champions' mood darkened further with the departure of Beram Kayal, Joe Ledley coming off the bench in his stead, before a powerful header from Efe Ambrose flew over the Inverness crossbar.
The visitors, showing plenty of vision and no little verve, extended their lead as Foran (35) latched onto Billy McKay's miscued overhead kick to nod the ball home from close range.
Mulgrew (42) offered the shell-shocked home support some succour when he beat Dean Brill form 14 yards having been picked out by Emilio Izaguirre clever cut-back.
A galvanised Celtic began the second-half on the front foot, Tony Watt slinging over a decent cross but Amido Balde could not direct his header on target.
Inverness were defending deeper although Doran did break at pace but, with the unmarked Nick Ross screaming for the ball, the former elected to go it alone, allowing the covering Virgil van Dijk to get back and avert the danger.
The pressure at the other end finally told when Scott Brown split the Caley defence with a superb pass to pick out the marauding Matthews (82) and the full-back worked some space before drilling home the equaliser.