Mattia Destro came off the bench on Sunday to score the winner with 23 minutes left in his first appearance of the season.
And despite having Miralem Pjanic sent off for a second yellow card in the dying stages, Roma broke their run of four consecutive draws to move within three points of Serie A leaders Juventus.
"Today what really mattered was to win and I am very happy for both the players and fans," Garcia told Sky Sport Italia.
"I wasn't worried by the run of four straight draws, because I was certain the victory would return eventually.
"It was also great for Destro, who had been out of action for a very long time. He did well, changed the game and made his mark on the afternoon.
"We put in a great performance with a lot of quality. (Goalkeeper) Morgan De Sanctis also made a decisive save, as he has a lot of experience and the mistake in Bergamo is behind him now."
Roma are still unbeaten in the league this season, with 11 wins and four draws from their 15 matches.
And Garcia is confident his side can cope with the absence of Pjanic - who will miss next week's clash with Milan due to suspension - and says his side are well and truly in the title race.
"I have a good squad and someone else will take his place," he said.
"Francesco Totti (might come in). (He) didn't have more than 15 minutes in his legs, so it was better to give him another week of intensive training and he can be ready next week.
"We are unbeaten and doing well, but Juventus are doing even better. They constructed a team to win the Scudetto and Champions League.
"I wish them all the best for Tuesday's game with Galatasaray in Istanbul, but Roma are in the Scudetto race too."
