The highest-ranked Norwegian in history was put through his paces over nearly three hours at Ken Rosewall Arena before downing his opponent 7-6 (7/6), 5-7, 6-4.
His win levelled the tie 1-1 after Karolina Muchova got the Czechs off to a flying start with a 6-2, 6-2 romp over Malene Helgo.
Machac and Muchova teamed up in the deciding mixed doubles with Ruud opting out after needing treatment on his left quad during the singles.
Instead, the Norwegian job fell to Viktor Durasovic and Ulrikke Eikeri and they lost 6-4, 6-4.
Ruud, who banked 2024 tournament wins at Geneva and Barcelona, said his match was a wake-up call.
"You kind of hope that it won't be that tough, but what a way to start the new season," he said.
"If every match is going to be this tough, it's going to be a tough (season)."
Ruud broke 25th-ranked Machac early in the third set to assume control, but he will be concerned at facing nine break points through the match.
World number 22 Muchova, the 2023 French Open finalist, was sharp from the first ball against Helgo, breaking serve four times in the 77-minute victory.
"It's the first match of the next year, and you never know how you'll come out and how you're going to feel," said Muchova. "I'm very happy to have won in two."
Helgo is still seeking her first United Cup win and is now 0-6 since her debut at the mixed-team tournament's first edition in 2023.
The 18-nation event is being played in Perth and Sydney.
Each team features three men and three women, with ties comprising one men's and one women's singles and an often decisive mixed-doubles clash.
Switzerland meet Italy in the evening Sydney tie.
Defending champions Germany, spearheaded by Alexander Zverev, face Brazil in Perth later followed by the Coco Gauff-led United States -- the 2023 winners -- taking on Canada.
