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 tie-break.
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 number one 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.
VENUS MARKS 90TH GRAND SLAM WITH OPENING WIN
Venus Williams marked her record 90th Grand Slam appearance with a three-set victory over Mihaela Buzarnescu of Romania on Tuesday.
The 41-year-old American, who is a five-time champion at the All England Club, triumphed 7-5, 4-6, 6-3.
Former world number one Williams first played Wimbledon in 1997 and is making her 23rd appearance at the tournament.
Now ranked at 111 in the world, she fired 10 aces and 33 winners to defeat her 160th-ranked opponent.
Up next is Tunisian 21st seed Ons Jabeur who eased past Rebecca Peterson of Sweden 6-2, 6-1. © AFP