
The 25-year-old top seed has never been beyond the fourth round at Wimbledon but two breaks of serve in the first set and then one at 4-4 in the second seemed to have set her up with a routine victory.
However, with errors creeping into Barty’s game Spaniard Suarez Navarro broke back and the set went into a tiebreak.
Suarez Navarro – who has only returned to play in recent months after battling cancer – dominated it and held five set points taking it with the first one 7-6 (7/1).
Barty, though, pulled herself together and played more like the world No 1 she is in the deciding set.
She showed little sign of the hip injury that forced her to retire from the French Open and made no mistake when holding three match points, taking victory with her first one.