The 25-year-old Dimitrov delighted Bulgarian fans with a barrage of sizzling winners, including some scorching passing shots. Dimitrov, who reached the Australian Open semi-finals last month, also fired down nine aces.
Basilashvili stunned top-seeded Dominic Thiem 6-4 6-4 on Thursday.
"I get excited, of course," Dimitrov, who closed out the match in 63 minutes, said in a courtside interview. "I'm happy but I have to stay focussed on the Sunday match."
Goffin stayed calm to fend off defending champion Roberto Bautista Agut 5-7 6-4 7-6 after the Spaniard took a 3-1 lead in the third set.
The Belgian, ranked 11th in the world, prevailed in two hours and 44 minutes of hard-fought tennis, sealing victory with a big serve that the defending champion returned long.
(Reporting by Angel Krasimirov, editing by Ed Osmond)
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