The Old Lady triumphed in their last-16 encounter thanks to Giovinco's well-taken strike, but no less than three players were ruled out of action in a worrying sight for the coach
Celebrations for the home team were muted, however, as they suffered injuries to Emanuele Giaccherini, Arturo Vidal and Nicklas Bendtner before and during the match that could stretch their squad to breaking point in the coming weeks.
Juve coach Antonio Conte was given a rousing reception by home fans as he returned to the bench in Turin for the first time following his suspension, and the hosts wasting no time in piling the pressure onto Cagliari. The first half was almost completely one-sided in terms of possession and territory, but the Sardinians were pardoned on more than one occasion by their rivals' imprecision in front of goal.
The lively Sebastian Giovinco was a constant thorn in the side of the Rossoblu, and after narrowly missing with an early free kick, the former Parma man was given an excellent chance to open the scoring. Giovinco read perfectly Leonardo Bonucci's lofted pass to break a faltering offside trap, but with the ball running away from him he could only release an underpowered shot that posed few problems for Cagliari keeper Avramov.
Bendtner also went close, flashing an effort past the post after fluid build-up from the Bianconeri. It was not to be the on-loan Arsenal man's night, though, as he struggled in vain to open his account for the Italian club.
Bonucci again inspired an opportunity with a well-hit cross as play broke down following a corner, but with no markers in his vicinity, Bendtner steered his header wide of the post. As half-time approached, the Dane broke down and was forced off, Alessandro Matri coming into the action.
Opportunities for the visitors were few and far between. Gabriele Perico briefly threatened Gianluigi Buffon's goal, but could only blaze over the bar after being played in by Sebastian Eriksson.
Juve finished the first 45 minutes well on top, but a pair of injuries left a sombre mood; as well as Bendtner, Giaccherini was forced out of contention during the warm-up, and his replacement Vidal was another casualty after pulling up with a muscle complaint.
Despite those setbacks, Conte's charges continued their quest to qualify for the quarter-finals of the Coppa, and Matri caused a stir when he headed Giovinco's free kick just wide of the net. The pressure was building, and with 10 minutes gone in the second half, Juve scored a decisive strike.
A low cross from the home team appeared simple to clear for Luca Rossettini, but the defender dawdled for far too long in getting rid of the ball. Juve took full advantage. Simone Padoin dispossessed his rival, and it fell to Giovinco, who jinked past the remaining markers to net the opening goal.
Having withstood the Serie A leaders for almost an hour, Cagliari would now have to drastically change their game plan, and come out for the first time to chase a result in Turin.
Daniele Dessena almost provided an instant response for the Sardinian institution, but the substitute saw his shot saved with class by the ever-reliable Buffon. Hopes were also briefly raised when Perico fired the ball into the far post, only to be extinguished once more as Mauricio Pinilla just failed to meet it with a despairing lunge.
With 15 minutes remaining, Cagliari were given perhaps their best chance of the match. Pablo Cepellini fired across the box following tenacious work from Victor Ibarbo in winning back possession, but his centre was diverted wide by Pinilla, who looked anguished at squandering the opportunity.
Juventus were in no mood to surrender their narrow advantage, and held on to book a place in the next round and keep their dreams of a third domestic double alive.
The Bianconeri become the third team to book their place in the quarter-final, joining Catania and Roma in the last eight. Five more ties are still to be decided, with Milan and Reggina continuing the round on Thursday evening.