Baltacha destroyed Olga Savchuk of Ukraine 6-3 6-0 in a match that lasted just over an hour but needed five hours to complete as thunderstorms interrupted play.
The 30-year-old Baltacha, a former British number one but now ranked 244th in the world, made light work of Savchuk, who is ranked 71 places higher than the woman born in Ukraine but raised in Britain after her footballer father joined Ipswich in 1988.
The other British woman left in qualifying, Sam Murray, was beaten 6-4 7-5 by rising American star Taylor Townsend.
The 17-year-old Townsend is ranked 144 places below Murray but she reached the final of the junior Wimbledon tournament and his highly rated as a star of the future.
Both remaining men did make it through.
World number 175 James Ward dropped the first set to 27th-seeded German Matthias Bachinger, but managed to win through 2-6 6-4 6-4. He will face India’s sixth seed Somdev Devvarman for a place in the main draw.
Davis Cup star Dan Evans, ranked 179th in the world, came through a topsy-turvy clash with Italian world number 159 Flavio Cipolla, winning 7-5 1-6 6-3 to set up a clash with Adrian Menendez-Garcia of Spain, who is ranked five places below Evans.