The 34-year-old Croat, who stunned third seed Agnieszka Radwanska in round two, came from a set down to end the dreams of Greek Maria Sakkari 3-6, 6-2, 6-3.
She will now play American qualifier, Jennifer Brady who shocked 14th seed Elena Vesnina 7-6 (7/4), 6-2.
It has been a memorable tournament so far for Lucic-Baroni, who won the Australian Open doubles title way back in 1998.
Until this year she had never gone beyond the second round since her debut in 1998, setting the new record of a 19-year gap between wins at a single Grand Slam tournament.
It surpassed the previous record held by Japan's Kimiko Date, who went 17 years between victories at Wimbledon between 1996 and 2013.