Raducanu led 3-1 with a break in the first set, but her sixth double fault to send it back on serve drew groans from her many supporters at Melbourne Park.
Worse was to follow.
Raducanu hit three more double faults in a poor ninth game, allowing Alexandrova to move ahead for the first time in the match at 5-4.
But the Russian could not serve out and paid the price as Raducanu played a superb first tie-break.
Raducanu appeared to have settled down in the second set, moving an early break up and confidently holding for a 3-2 lead.
Then the double faults reappeared.
Three more in the sixth game heaped on the pressure, but Raducanu scrambled to save two break points before Alexandrova did take her serve for 4-4.
After failing to serve out at 5-4, Raducanu saved a set point in the 12th game with a first serve which Alexandrova – an aggressive ball-striker who cut a demonstrative figure – whacked a backhand long.
Raducanu again dominated the tie-break, racing 6-0 ahead and clinched victory on her third match point – after hitting a double fault on the first.
“That was the cherry on the top,” Raducanu laughed.
On stepping up in the tie-breaks, she added: “I think that is one of my strengths.
“When it gets to those clutch moments, I relish playing in that.”
