The third-seeded Davydenko, who missed the Australian Open with a foot injury, needed just over an hour to dispatch Andreev.
Two other Russians, Dmitry Tursunov and Mikhail Youzhny, also advanced by beating two Frenchmen. Tursunov beat Arnaud Clement 6-0, 6-3 and Youzhny overcame Laurent Recouderc 6-4, 6-4.
"The first round is always difficult,'' the 26-year-old Tursunov said. "You always have to find your rhythm and get accustomed to a new surface.''
Julien Benneteau also advanced, defeating Feliciano Lopez 6-4, 5-7, 6-2. Andreas Seppi of Italy ousted Dutchman Thiemo de Bakker 6-7 (5), 6-2, 7-6 (1).
Top-ranked Rafael Nadal was set to play his first singles match since winning the Australian Open on Wednesday against Simone Bolelli, and second-seeded Andy Murray will play Ivan Ljubicic on Tuesday.
"It's going to be a very tough match because Ljubicic is a good indoors player.'' Murray said.
Capello Satisfied With Spain Test
Tursunov wins opening match at ABN Amro
