
The American needed six match points to prevail after saving Luczak's three chances to take the tight match into a third and deciding set.
Defending Australian Open champion Serena Williams had guaranteed the four-time champion Americans a place in Friday's title showdown with it all to play for in the opposing group for Serbia, France and Taiwan.
Only Argentina is out of the trophy chase at the eight-nation team event.
Williams, winner of eight Grand Slam titles, started the day as she beat Australian Alicia Molik 6-2, 7-6 (9-7), making it impossible for the hosts to reach the last day by eliminating the possibility of a 3-0 Australian sweep in the tie.
The United States won its last title at the tournament in 2003 when Williams had teamed up with James Blake.
Williams said she remained motivated by the diamond-studded tennis balls given to each of the trophy-winning players. The American added that her serve made the difference against Molik.
"My serve was working really well," she said. "I'm so proud of it. It's really exciting. I hope I can keep serving like this over the next weeks (of the Australian Open)."
Williams advanced in 85 minutes, breaking Molik twice in the first set and winning after the Australian saved a first match point in the concluding game.
Fish had more of a battle against Luczak, who is aiming to crack a top 50 ranking this season.
Fish produced huge serves almost on demand, firing aces to save set points for Luczak in their second-set tiebreaker.
But conversion on two of seven break points and 41 errors showed that the Floridian could do with sharpening his game before the final.
"It was a great match - those tiebreakers could have gone either," Fish said. "I could have lost in straight sets. Peter really makes you work."
Group A of the event was being decided later as Serbia faced Taiwan and France played Argentina.
"Serbia (with the world number three pair of Jelena Jankovic and Novak Djokovic) is the obvious favourite," Fish said. dpa