Juve counterpart Antonio Conte, meanwhile, took the opposite approach.
He rested goalkeeper Gianluigi Buffon, a number of midfield stars including Paul Pogba, Kwadwo Asamoah and Stephan Lichsteiner and strikers Carlos Tevez and Fernando Llorente.
Roma, as might be expected, dominated the early exchanges with Radja Nainggolan and Alessandro Florenzi seeing shots go wide, but it took until late in the half for the Giallorossi to threaten.
Vasilis Torosidis whipped in a superb curling ball at the far post with which Kevin Strootman, at full stretch, narrowly failed to make contact.
Juventus had the ball in the net shortly after the restart when Federico Peloso headed past Juve number two 'keeper Marco Storari from close range, but his effort was ruled offside.
Florenzi was next to come close with a header from Torosidis which took Storari by surprise but the second half was nowhere near as productive as the first.
As the game stalled, Garcia replaced Florenzi with Miralem Pjanic while Conte attempted to waken his team from their relative slumber by replacing striker Sebastian Giovinco with Llorente on 77 minutes.
Garcia's switch was to pay dividends, Pjanic reaching the ball before Andrea Barzagli to stride forward and set up Strootman deep on the left.
The Dutchman's cross arrived in the box and Gervinho did well to tap the ball home while in mid-air with the side of his right boot in the 79th minute.
Roma were fired up and Totti delivered for Gervinho which the striker missed, and minutes later the Ivorian sent a low, weak shot into the arms of Storari after late substitute Adem Ljajic had carried the ball from midfield.
Pjanic then came close with a shot in injury time, while Juve's frustrations boiled over with Chilean Arturo Vidal cautioned for dissent.
On Wednesday Milan host Udinese for their second match under new coach Clarence Seedorf with high-flying Fiorentina away to Serie B side Siena a day later.
Napoli host holders Lazio in the last quarter-final on 29 January.