The Ivory Coast forward, who has been widely criticised for poor finishing, met Kevin Strootman's cross with an acrobatic finish 11 minutes from time to win a niggly tie.
"It was a win that we wanted at all costs," said Roma captain Francesco Totti. "We had to get our own back for the horrible defeat we suffered in Turin. Today we showed ourselves to be at the same level as a very good Juve side."
Juve coach Antonio Conte Added: "It was a very even match between two very strong sides. It's upsetting having conceded the goal in our best moment of the match."
In the two-legged semi-finals, the Giallorossi will face one of their deadly rivals, Napoli or Lazio, who meet at the San Paolo Stadium in Naples next Wednesday.
The win means Roma remain on track for a 10th Coppa Italia triumph and gives them a psychological boost in the title race, although Juve started without key players including midfielder Paul Pogba and forwards Fernando Llorente and Carlos Tevez.
"For me there isn't a big gap between us, even if it's the league table that counts. Eight points is a lot, we're having a brilliant season, but they're doing even better," added Totti.
"If we carry on the way we played in the first half of the season we'll be close to Juve, but if they stay at the same level, it will be difficult."
Juventus used the same tactics that saw them win the league clash, defending deep and allowing Roma to dominate possession.
However this time they were not allowed to cause any damage on the break, with their only real scoring chance coming when Federico Peluso's header a minute after the break was disallowed after the linesman ruled that the cross had gone out of play.
"Peluso's goal should have stood. For Isla's cross the ball was in and not only just either," addede Conte.
After Gervinho sent the home fans wild with his karate-kick winner, the visitors could not break down a brilliantly drilled Roma side who have four games in a row in league and cup since losing at the Juventus Stadium straight after the winter break.
In the other quarter-finals, AC Milan will look to continue their winning start under their former player Clarence Seedorf against Udinese on Wednesday, while Siena travel to Fiorentina for a Tuscan derby on Thursday.