Robson put in an impressive performance throughout, serving well in the first set to stay one step ahead of Garica.
The Brit hit two consecutive aces to edge in front at 4-3 before Garica levelled on her serve.
However, Robson got the crucial break, forcing her opponent to hit a long forehand. She soon wrapped up the first set 6-4.
The second set went with serve, until at 5-5, Garcia broke when Robson hit a double fault.
Serving for the set, Garcia chalked up two set points but double-faulted the first before Robson saved the second with a backhand.
She then broke back to level the set at 6-6.
Into the tie-break and Robson brought up four match points and secured the set and victory with the fourth to win 6-4 7-6.
She will face Li Na in the next round.