Wimbledon champions Cash and Glasspool beat Norwegian pair Nicolai Budkov Kjaer and Viktor Durasovic 6-2, 2-6, 7-6 (5) to send Britain through to the next round in September.
"It means the world to get our first point and win the tie for GB," said Cash.
"We were ready for everything. We never take for granted that a match is going to be easy, and we're happy to get it across the line."
Britain surged into a 2-0 lead on Thursday as Jack Draper made a triumphant return from an arm injury before Cameron Norrie won the second singles rubber in Oslo.
Hosts Norway were without their top player, Casper Ruud, the world number 12, who pulled out of the tie to spend time with his partner and newborn child.
"We've done a really good job and played some really good tennis," said Draper. "Bit different with Casper not playing, but we've done our job."
Britain will face either Australia or Ecuador for a place in November's eight-team Davis Cup Finals.
In Duesseldorf, the qualifying second seeds Germany, without world number three Alexander Zverev, took a 2-0 lead over Peru with two straight-set singles wins.
Yannick Hanfmann beat Gonzalo Bueno 6-4, 6-4, and then Jan-Lennard Struff dispatched Juan Pablo Varillas 6-4, 6-2.
Croatia and Denmark were level 1-1 in Varazdin after the opening singles, and Japan and Austria were also all-square in Tokyo.
Elsewhere this weekend, Serbia and Chile go head-to-head in Santiago in a bid to set up a tie with Spain.
Ten-time winners France take on Slovakia in a tie starting Saturday, and Belgium, surprise semi-finalists last year, play Bulgaria in Plovdiv.
