Antonio Conte's side cruised to victory on Wednesday night thanks to three first-half goals and will face the Viola in the last eight.
The Bianconeri opened the scoring after just seven minutes through Sebastian Giovinco's wonderful, curling effort.
The forward controlled a pass from Fabio Quagliarella on the edge of the box before shifting the ball onto his right foot and dispatching a fine effort into the far corner.
It didn't take long for the Serie A champions to double their advantage. Six minutes later it was 2-0 to the home side, with Giovinco again at the heart of the goal.
The Italy international delivered a fine free-kick into the penalty area which found the head of the unmarked Martin Caceres and the defender was left with the easiest of finishes.
Juve was all but out of sight and with just nine minutes left in the opening period. Another brilliant free-kick from Giovinco caused all sorts of problems in the visitors' box and this time Quagliarella was on hand to prod the ball home at the near post to make it 3-0.
Juve's dominance continued in the second half as Avellino goalkeeper Giuseppe Di Masi just about held on to Quagliarella's powerful free-kick, while Stephan Lichsteiner should have scored when he steered wide one-on-one in injury-time.
Antonio Conte's side couldn't add to its tally before the referee blew his full-time whistle, but its first half efforts proved enough and the Viola now await in the last eight of the Coppa Italia.