Bogdanovic is the world number 161
British number two Alex Bogdanovic is two victories away from reaching the main draw at Wimbledon after winning his first qualifying match on Monday.
The 26-year-old beat Brazil's Ricardo Hocevar 6-3 7-6 (7-3), breaking in the eighth game of the first set before winning the second-set tie-break.
Bogdanovic did not receive a Wimbledon wildcard this year having been handed one for the last eight tournaments.
Britons Dan Evans and Josh Goodall also won their matches at Roehampton.
British number three Evans fought back from a set down to beat American Tim Smyczek 4-6 6-1 12-10, while Goodall beat 21st seed Albert Ramos-Vinolas 6-3 6-4.
World number 161 Bogdanovic, who was not recommended for a wildcard despite being inside the Lawn Tennis Association's guideline cut-off point of 250 in the world, now faces former Queen's finalist Nicolas Mahut.
Evans will have to overcome Lithuania's Ricardas Berankis, while Goodall will meet Russian Evgeny Kirillov.
There was no joy for Surrey player Chris Eaton, who reached the second round at Wimbledon in 2008. He was beaten 4-6 6-3 10-8 by Croatia's Roko Karanusic.
Fellow Britons Morgan Phillips, Daniel Cox, David Rice, Josh Milton and Alex Ward also exited at the first qualifying stage.
Meanwhile, compatriot Dan Smethurst retired after losing the first set 6-0 to Ryler Deheart.