Gervinho scored the decisive goal in the 79th minutes to seal a 1-0 victory and advance to the semi-finals.
Roma will play Napoli or Lazio in the semis, which will be decided across two legs.
"No, I don't think we are the favourites. There are still great teams in the competition and the two semi-finals will be at a very high level," Garcia told RAI Sport.
"We should enjoy this victory and from tomorrow be concentrated on Verona, where we'll play with great ambition.
"Napoli or Lazio makes no difference to us."
Roma trail Serie A leaders Juventus by eight points and were recently humbled 3-0 by the two-time defending champions, but Garcia hopes this victory will give his side some momentum into the second half of the season.
"Tonight was the Coppa Italia, but it was also the Serie A leaders against second, so it counted for the environment," he said.
"We saw a Roma side with character who took the initiative in Turin, but this time we created spaces and were able to strike at the right moment.
"We are competitive, but also know Juve are a great side who won two consecutive Scudetti. We did not expect Juve to have 55 points after 20 games, as 47 is an exceptional tally already.
"Tonight we saw we have the abilities and the talent to beat a side that is one of the biggest clubs in Italy.
"How strong are Roma? We'll see at the end of the season, but we have to keep pushing and see where we end up. I am truly satisfied."
