Roma - runners-up last season - were patient throughout against a Fiorentina side renowned for their possession-based football, with Radja Nainggolan and Gervinho on target at the Stadio Olimpico.
Nainggolan got the ball rolling in the 28th minute, pouncing on Gervinho's effort that had been parried into his path.
The Rome-based outfit finally sealed the points in the 90th minute after Gervinho rounded Fiorentina goalkeeper Neto.
"What counts is that at the end we scored the second goal," Garcia told Sky Sport Italia.
"It certainly would've been better if it had arrived earlier, but the team played well, especially in the first half.
"Let's not forget we were playing against Fiorentina, who specialise in possession, yet we kept the ball for the first hour."
Garcia added: "The players worked hard when attacking and defending. We were aggressive in the first half and won back the ball quickly.
"With Fiorentina's diamond midfield you need to be patient and pass the ball around to find space, so I asked the team for patience.
"I saw Roma confident in the first hour and in control of the game."
Meanwhile, midfielder Daniele De Rossi said Roma are slowly finding their feet after an influx of new arrivals that includes Greece international Jose Holebas, whose signing was confirmed following the match on Saturday.
"We are a strong team and we are growing," De Rossi told Sport Mediaset.
"We suffered [tonight] against one of the best teams in the league. The first game always has its ways, but we have an organised team that will help us acclimatise to the new players better."
