Johansson, the third seed, posted consecutive wins for the first time since the Tennis Masters Monte Carlo in April as he eliminated Croatia's Ivan Ljubicic, 6-3, 6-4.
"I'm so happy with my performance today," Johansson said. "I played really solid. I served and returned really well."
Kafelnikov, the top seed who is ranked just 34th in the ATP Champions race, notched a 6-3, 7-5 win over French qualifier Cyril Saulnier.
In the quarterfinals for the first time since the Pacific Lite Open in March, Kafelnikov improved to 19-18 overall in 2002.
"It's definitely different than before, but I'm not sure whether I can judge my performance on the match today," Kafelnikov said. "To win any matches for me at this stage is important. I'm only lacking confidence and by winning matches I will get it back."
Bohdan Ulihrach of the Czech Republic also moved into the quarterfinals with a 6-2, 6-7 (2-7), 7-5 win over German Rainer Schuettler.
Michel Kratochvil had a 6-3 lead over Denmark's Kenneth Carlsen when rain forced the postponement of their second-round match. It will be completed Thursday.