Two goals in as many minutes from Daniele De Rossi and Alessandro Florenzi sealed the points at the Stadio Armando Picchi on Sunday for the capital club, who finished sixth in Serie A last term and lost the Coppa Italia final to rivals Lazio - missing out on European football in the process.
But the former Lille Garcia manager is taking things as they come, while hailing his goalscorers, especially Italy Under-21 international Florenzi, whom Garcia believes is the biggest attacking threat in his side.
The Frenchman said: "The team is calm. Last season is over, we've begun a new one and we've moved on.
"I'm happy, we didn't concede, we scored two and it's almost fate that Daniele De Rossi found the net.
"We played like a team and had patience in a tight match like this. These are three vital points to begin the campaign well.
"Florenzi? He's an important player, he's able to play the ball first time, he runs a lot for the team and he's our top scorer at this moment in time."
Garcia has been busy in the transfer market during the close-season, bolstering his ranks with 14 signings to counteract the departures of Maarten Stekelenburg, Marquinhos and Pablo Daniel Osvaldo.
But the 49-year-old has no immediate plans to bring more players to the Stadio Olimpico, and wants to focus on affairs on the pitch.
"We're not thinking about that," he added. "We'll enjoy this victory because it's always tough to win away."
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