Retiring two-time grand slam champion Lleyton Hewitt will play Aleksandr Nedovyesov in the final singles clash to decide which team goes through to a semi-final against either Britain or France.
World number 68 Groth, who teamed up with Hewitt to win the doubles on Saturday and keep Australia in the tie, kept his composure in the decisive tiebreak to seal the grass-court match 6-3 7-6(4) 4-6 7-6(6) and level the tie 2-2.
After clinching it when Kukushkin clubbed a shot into the tramlines, Groth threw his racket way, hurled his cap behind his head and pumped his fists like a boxer as local fans roared their approval on a glorious day in the tropical city.
"We've still got one more to go but this is probably the most amazing feeling I've had in tennis," an ecstatic Groth said in a courtside interview.
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"This is unbelievable."
(Reporting by Ian Ransom; Editing by Peter Rutherford)
