Meyer's 6-2, 7-6 (9-7) decision over Andreev gave the Germans an insurmountable 2-0 advantage against Russia in the tie, garnering Germany a 1-0 record in the round-robin tourney's Blue Group.
Germany completed the triumph when the doubles team of Mayer and Christopher Kas bested Russia's Igor Kunitsyn and Dimitry Tursunov 6-4, 3-6, 10-8.
The Czech Republic also took a point Monday, beating Japan 2-1 in the Red Group with the help of a doubles victory by Radek Stepanek and Frantisek Cermak, who dispatched the Japanese tandem of Tatsuma Ito and Go Soeda 6-2, 7-6 (7-4).
Serbia, meanwhile, went to 1-0 in the Blue Group by besting Croatia 2-1 thanks to Janko Tipsarevic's 6-0, 6-0 rout of Lovro Zovko.
Argentina, which defeated the United States Sunday, finished off the victory with a 7-6 (7-2), 6-4 doubles win Monday.