Sergio Oliveira was the hero for the visitors as he struck twice including an 115th-minute free-kick.
This sent his side through after 120 minutes of scintillating football.
Juventus got off to the worst possible start, with Merih Demiral giving away a penalty within the opening 20 minutes.
Sergio Oliveira gave Wojciech Szczesny no chance as he tucked his spot kick away.
And suddenly, Juve needed to score twice just to force extra-time.
The Italians were unable to get on the scoreboard in the opening period with Porto going into half-time in a comfortable position.
But Chiesa blew the tie wide open within four minutes of the restart.
The winger thundered home a right-footed strike from a neat Cristiano Ronaldo pass.
After smacking the post, Chiesa was the hero again as he levelled the tie with a little under half an hour to play.
However, with five minutes to play, the midfielder unleashed an unstoppable free-kick to give his side a 4-3 aggregate lead.
Juventus will now be forced to wait at least another year in their bid to win a first Champions League crown since 1996.
Ronaldo’s £99million arrival in 2018 was supposed to bring the glory days back to Turin.
But at the age of 36 and the club either keeping Ronaldo and his sky-high wages or selling prospect Matthijs de Ligt.
Juve have a huge decision to make this coming summer.