Kwadwo Asamoah's well-struck goal for Juve with 10 minutes remaining cancelled out Kevin Mirallas' impressive individual effort just past the hour mark.
The game went straight to penalties after 90 minutes, with Leon Osman missing his spot-kick, before Andrea Pirlo did likewise for the Italian side.
Everton goalkeeper Tim Howard then saved from Federico Peluso in the sudden-death shoot-out to give the Toffees a 6-5 spot-kick success.
Earlier, Roberto Martinez's team had an early Steven Pienaar free-kick disallowed for offside, while John Stones hit the post with a header just before half-time.
The Italian champions included Carlos Tevez, a summer signing from Manchester City, in their side.
Source: PA
Source: PA