To keep their hopes of a quarter-final place alive, Team GB had to defeat Moldova 3-0 and hope that the other results in Group C in Sydney went their way.
The first of those objectives was achieved in quick time with all three rubbers requiring just over the hour to complete.
In the opening rubber Cam Norrie beat Alexander Cozbinov 6-2 6-2 in just 71-minutes while Dan Evans required one minute more to outclass Radu Albot ranked four places below him, by the same scoreline.
Even Jamie Murray, in harness with Joe Salisbury, just failed to match the score line with the pair going through against Albot and Cozbinov 6-2 6-3 in 62-minutes.
Team GB have completed the round robin stage with two wins and one loss and their fate is now and depends on the outcome of the evening session clash between Bulgaria, favourites to top the group with two wins to their credit, and Belgium with just the one.
The top nations from the six groups at the inaugural 24-team event progress to the quarter-finals along with the best two runners-up.
Commenting after their quick-fire win, captain Tim Henman said: “I would say my concern for the Evans match was how he would respond after putting in such a good performance against Goffin. It’s not always easy to back that up.
“I mean, he was world class again. Gave so little away. Got off to a very good start, took his opportunities, and it was a really clinical performance.”
Evans meanwhile, said he was enjoying the experience of starting the season with a team event, with everyone supporting each other.
“I mean, it’s enjoyable to have the guys on the side. To (have them) sort of come out every point, point in, point out, has really helped. Not just me. I think everyone is sort of focused a little better than on tour.
“So I don’t think it’s an accident that I have been playing good, focusing on that. I’ll definitely use that for when I’m carrying on during the year.”
Britain’s top player faced Albot in last year’s Delray Beach final, arguably the biggest match of their careers. Despite holding three championship points, Evans fell short, with Albot lifting his maiden ATP Tour trophy.
“I have known the tennis he’s capable of. I would just say the challenge is to deliver that consistently throughout a match, consistently throughout the tournament and throughout a year,” Henman, a former world No 4, said of Evans.
“And I think the element that we focused on is really the mental side of things and just playing it point by point, trying to structure the points on his terms so he can be aggressive and dictate at the right moments.
“He’s done a great job of that.”
In Brisbane, Canada beat Germany 2-1 and will likely advance to the quarterfinals as one of the two best second-place finishers in group play.
Alexander Zverev’s terrible run of form continued as he surrendered meekly to Denis Shapovalov 6-2 6-2 in just 70-minutes to bring the Canadians back into the tie following Jan-Lennard Struff 6-1 6-4 win over Felix Auger-Aliassime to give Germany the early lead.
The tie was then clinched by Auger-Aliassime and Shapovalov 6-3 7-6(4) over Kevin Krawietz and Andreas Mies in a high quality rubber.
Zverev needed to beat Shapovalov to keep Germany in the hunt for a quarter-final spot in Sydney later this week.
But Shapovalov took control from the outset against an opponent whose serve was misfiring and whose body language suggested a player badly out of form, with less than two weeks before he appears at the Australian Open.
“Generally I’m not playing good, so there’s a lot of things that I still need to improve, but it’s the start of the season,” Zverev said.
“The first match I actually thought I played okay — the serving was horrible — but then I played two very bad matches, but what can I do?”
At Perth, Russia clinched its match with Norway with wins in both singles to top Group D. Karen Khachanov beat Norway’s Victor Durasovic 6-2 6-1 in the opening singles match and Daniil Medvedev beat Casper Ruud 6-3 7-6(6) on his second match point with an ace.
Teymuraz Gabashvili and Konstantin Kravchuk finished off a 3-0 clean sweep of the tie by winning the doubles against Casper Ruud and Viktor Durasovic 7-6(4) 6-4.