The capital club were defeated 2-0 at Stadio Olimpico by last season's runners-up thanks to two second-half goals from Mario Gomez, who was in clinical form.
Roma went close to taking the lead several times in the first half, as Alessandro Florenzi, Radja Nainggolan and Adem Ljajic all wasted good opportunities.
Garcia left frustrated by his team's inability to find the net, but also conceded that a lack of options was proving problematic.
"We played well in the first half but it wasn't enough to win the game," said the Frenchman.
"They [Fiorentina] scored with their first chance but that's football. Being clinical is important, especially in a knockout tie.
"We're conceding too many goals at the moment and we're not as sharp up top. We have fewer options.
"I had to throw on [new signing Victor] Ibarbo who hadn't played for six weeks and had only trained once. He's the positive from today.
"We need to be better defensively and more clinical going forward, [but] we'll get straight back to work and we still have Serie A and the Europa League to play for.
"We have to win at Cagliari [in Serie A on Sunday] to get luck back on our side. Then our absentees will return and we'll be stronger.
"We've had a number of injury problems I didn't expect. Perhaps we needed a little more rub of the green.
"You see a club's strength when the going gets tough. We'll all fight, me first and foremost, to win the next one."
Despite such a disappointing result at home, greeted by boos at the final whistle, Garcia marvelled at the fans' treatment of the team, thanking them for their "fantastic" support during the match.
"I want to stress that the Curva Sud [fans in the southern section of the Stadio Olimpico] got behind the team throughout the match. It was fantastic.
"The fans are disappointed as are we. But rest assured we'll get our heads down to try and return to winning ways."
