The Giallorossi eliminated Juventus on Tuesday, but the coach is adamant that his side should not be seen as superior to the other teams left
Gervinho scored the decisive goal in the 79th minutes to seal a 1-0 victory and advance to the semi-finals, where tthey will play Napoli or Lazio.
"No, I don't think we are the favourites," Garcia told Rai Sport. "There are still great teams in the competition and the two semi-finals will be at a very high level.
"We should enjoy this victory and from tomorrow be concentrated on Hellas Verona, where we'll play with great ambition.
"Napoli or Lazio, it 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," the former Lille boss added.
"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.