Jose Callejon, Gonzalo Higuain and Jorginho struck in Napoli's 3-0 second-leg win on Wednesday, giving them a 5-3 aggregate victory.
Roma had led 2-0 in the first leg but needed a late winner to secure a 3-2 victory, and Garcia believes it was that match which proved costly.
"Perhaps we lost the qualification in the first leg, as we conceded two goals when we were 2-0 up," Garcia told a news conference.
"This evening we had three chances to score in the first half and didn't make the most of them. We should've been more ruthless in attack and also in defence, as we made mistakes in both legs."
Callejon opened the scoring on 33 minutes when he nodded in Christian Maggio's well-weighted cross from the right.
Higuain headed in Napoli's second in the 48th minute before Jorginho was played in by Dries Mertens and finished in a one-on-one.
Garcia felt his team should have held a half-time lead, but was full of praise for Napoli, who will face Fiorentina in the final.
"If there was a side that should've been in front at half-time, then it was us. The fact we weren't was our fault," he said.
"The start of the second half was not good and with two quick goals we lost the tie. We've got to improve. When it was still 1-0 to Napoli, it was wide open and anything could've happened.
"I compliment Napoli, they have a strong attack and I wish them all the best for the final.
"As for Roma, we've got to think positive because we do create chances in every game. The key to the future is to be more effective up front.
"Now we close the Coppa Italia book and re-open the Serie A one. We've got to win our next game against Sampdoria.
"For the moment we are four points ahead of Napoli in the league and have a game in hand."
