World number five Andreeva, playing her first match on grass in 2026, pulled out of last week's Berlin Open on the surface.
The Russian had said that she wanted more time to recover and prepare for Wimbledon after her June victory on the Roland Garros clay.
The 19-year-old matched her compatriot early in the opening set before losing seven straight games in hot conditions in central Germany.
Andreeva was down 4-0 and 40-15 in the second set but fought off the break point and won three straight games to elbow her way back into the match.
But 31-year-old Alexandrova, ranked 19th in the world, steadied herself to serve out the set and win the match in one hour and 28 minutes.
Alexandrova will face Japan's Naomi Osaka in the quarter-finals.
Also on Wednesday, Romanian Elena-Gabriela Ruse defeated Russian Anna Kalinskaya 7-5, 6-2 while Dane Clara Tauson beat China's Zheng Qinwen 5-7, 6-4, 6-2.
Later on Wednesday, defending Wimbledon champion Iga Swiatek, a finalist at Bad Homburg in 2025, takes on Emma Navarro.