The Giallorossi conceded two controversial penalties, with referee Gianluca Rocchi pointing to the spot when an Andrea Pirlo free kick struck Maicon's arm on the edge of the box before ruling that Miralem Pjanic had tripped Paul Pogba on the left-hand side of the area right on half-time.
Carlos Tevez converted on both occasions and Leonardo Bonucci then netted the game's deciding goal with just four minutes remaining.
Garcia, who was sent to the stands following the second penalty, was proud of his players at the full-time whistle - but less impressed with the officials.
"It's a pity that here the penalty area is 17 metres," the Frenchman told Sky Sport Italia. "But I'm happy with my team, who showed great personality.
"There were many incidents today but it was also a little bit our fault that we lost.
"We missed two great chances to score and we were beaten by three set-pieces.
"We could have done better, not only in the result but also our play."
The victory sees Juve move three points clear of Roma at the top of Serie A after six rounds of action.