Britain and Croatia booked spots in the Davis Cup quarter-finals after topping their respective groups on Sunday, while 28-times champions Australia were knocked out.
Britain's doubles pair of Joe Salisbury and Skupski won in straight sets to seal the tie against the Czech Republic 2-1 and secure top spot ahead of France.
Britain initially made things difficult when Dan Evans lost his singles rubber against Tomas Machac, capitulating in the second set when he looked on track to force a decider.
The Briton, ranked 25 in the world, won four games in a row and then went 5-2 up in the second set before losing five straight games to give 143rd ranked Machac a 6-2 7-5 victory.
However, British number one Cameron Norrie managed to pull one rubber back after being stretched to three sets, beating Jiri Lehecka 6-1 2-6 6-1.
"It's never easy being favourite and never easy in this situation but I really enjoyed the match," Norrie said. "All credit to him but I managed to come back for a must-win match."
Croatia, who beat Australia 3-0 in their opening tie, booked top spot in their group and a quarter-final tie against Italy after Nino Serdarusic beat Hungary's Fabian Marozsan 6-4 6-4.
The 2018 champions were pegged back, however, when Hungary's 282nd-ranked Zsombor Piros recovered to beat former US Open champion Marin Cilic 4-6 7-5 6-4 - his second major scalp after dispatching Australia's John Millman on Saturday.
Croatia won the doubles decider and the result means Australia finished runners-up but cannot advance as one of two best second-placed teams with a 2-4 win-loss record.
Kazakhstan also reached the last eight as group winners after Mikhail Kukushkin and Alexander Bublik won their singles rubbers against Canada to take an unassailable 2-0 lead, with Sweden finishing second in the group.