Sharapova, the reigning Australian Open champion, overpowered Israel's top player, Shahar Peer, 6-1, 6-1. Sharapova did need to save two break points before she closed out Peer in the third set.
Chakvetadze had no problem beating Tzipora Obziler 6-4 6-2.
"It's a great win, although it is strange to win a match without actually feeling like you won," Sharapova said. "But this my Fed cup debut and I'm very satisfied with my performance."
Russia next faces the United States-Germany winner in one semifinal.
Chakvetadze replaced Dinara Safina in Sunday's reverse singles.
In other quarterfinals, China edged France 3-2 in Beijing and Spain eliminated Italy by building an insurmountable 3-0 lead in Naples.
China meets Spain in the other semifinal.
2014: The Year in Music [PHOTOS]