Barthel's tournament came to a premature end as American Brengle scrapped to a 6-7 (6-8) 7-5 7-6 (7-2) victory in the first meeting between the two on Monday.
Both players were broken twice in the first set, but it was Germany's Barthel who went in front by winning a tie-break which she edged by securing five mini-breaks to Brengle's four.
Brengle, ranked 84th in the world, looked to be up against it when she trailed 4-1 in the second set after being broken twice, but showed character to win six of the next seven games and take the contest into a deciding set.
World number 40 Barthel was on the brink of the second round when she led 5-2, but Brengle again rallied to force another tie-break and used that momentum to secure her progression by claiming three mini-breaks to Barthel's one.
Kateryna Kozlova was another qualifier to reach the second round on Monday, beating Kirsten Flipkens 4-6 6-3 6-4.
Ukrainian world number 138 Kozlova saved nine break points and secured four breaks of her own to knock out Belgian Flipkens.
Kozlova saved two break points in what proved to be the final game of the match before taking the first match point she earned to account for the world number 50.
Top seed and home hope Casey Dellacqua overcame a second-set wobble to beat American Lauren Davis 6-1 4-6 6-4, while second seed Zarina Diyas was a 6-3 6-4 winner against Silvia Soler-Espinosa.
Italian Camila Giorgi was also a first-round winner on Monday at the expense of Australian wildcard Storm Sanders, who was beaten 5-7 6-1 7-6 (8-6).
Giorgi's compatriot Roberta Vinci - the ninth seed - also beat an Australian opponent in the form of Olivia Rogowska, winning 6-1 4-6 6-3.
Daniela Hantuchova began her competitive comeback from knee strife with a hard-fought 7-6 (12-10) 7-6 (7-5) victory over qualifier Richel Hogenkamp.
Karin Knapp, Annika Beck and Heather Watson are through to the second round, while Johanna Larsson advanced after Zheng Saisai was forced to retire and eighth seed Alison Riske went through when Christina McHale was unable to play on after the first set.
Mirjana Lucic-Baroni also cut short her tournament when she retired while trailing Bojana Jovanski 5-2 in the first set.