Ward (pictured) recovered from a set down to beat Mischa Zverev of Germany 6-7(1) 6-4 6-4 to secure his progression, while Baker overcame Italian 19th seed Matteo Viola in three sets, 3-6 6-2 6-4.
Baker takes on American Donald Young as he eyes a place in the main draw of the first Grand Slam of the year.
The 26-year-old has met Young just once before, recording a straight-sets win at Eastbourne last year.
Ward, meanwhile, takes on Germany’s Julian Reister in a first-time meeting for the pair in the final round of qualifying.
It was less good news for Johanna Konta, however, as the world number 153 suffered a 7-5 2-6 8-6 defeat to Yi-Miao Zhou of China in the second round of qualifying for the women’s main draw.
British number one Andy Murray will begin his bid for a second Grand Slam title against Dutchman Robin Haase, with a potential semi-final meeting with Roger Federer on the horizon.
Laura Robson meets the 2012 Birmingham champion Melanie Oudin in the first round at Melbourne Park, while Heather Watson faces Romania’s Alexandra Cadantu.
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