Brazilian Costa opened after quarter of an hour in the Allianz Stadium with Croatian Mandzukic adding the second amid controversy on 69 minutes to set up a meeting with Atalanta, who eliminated Serie A leaders Napoli 2-1 on Tuesday, in the final four.
Lazio, finalists in 2017, will meet AC Milan in the other semi-final played over two legs on January 31 and February 28th.The winning team will earn a direct entry to the group stages of the Europa League.
"We've reached the semi-final for the fourth year running and the lads took the right approach to this game," said coach Massimiliano Allegri.
Torino had wanted a free-kick for Sami Khedira's challenge on Afriyie Acquah in the build-up to Mandzukic's goal which was awarded after consultation with the video assistant referee (VAR).
"I didn't see the incident and I will not comment. I never comment on VAR. I will just say we could have scored another couple of goals at least."
Allegri added that the three-time reigning champions were not taking anything for granted against Atalanta who won their only ever title in the Italian Cup back in 1963.
"Atalanta deservedly beat Napoli to reach the semi-final and it won't be easy against them, but we'll make sure we are ready," added the Juventus coach.
Torino suffered badly during their last meeting with Juventus 4-0 in September but this time came close to scoring through Mbaye Niange just after Costa's opener.
But they were no match for the record 12-time winners with Costa almost getting a second after 59 minutes with Paulo Dybala also on form.
Mandzukic beat the offside trap to lob over Vanja Milinkovic-Savic in the Torino goal, with VAR confirming to the fury of coach Sinisa Mihajlovic who was banished from the sidelines for dissent.
Torino assistant manager Attilio Lombardo blasted a decision "which killed off the game".
"Acquah gets the ball and is then hit by Khedira. I don't understand what is evaluated via VAR and I don't understand how the referee saw the opposite," said Lombardo.
Torino -- who eliminated AS Roma in the last 16 -- now turn their focus back to Serie A to try to earn European football next season as they sit tenth.
"There are regrets, it was a good opportunity, but Saturday against Bologna will determine a lot of our journey."