Poland and Italy will face off in the other semi-final on Monday.
Raducanu broke in the first game as she started off with a bagel against Rebecca Marino, who made her debut in the event in 2011.
The second set went with serve until the 2021 US Open champion broke to lead 6-5 before wrapping up a 6-0 7-5 victory to give her team a 1-0 lead.
Emma Raducanu beat veteran Rebecca Marino in straight sets. (AP PHOTO)
Boulter then sealed their last-four spot as she beat Leylah Fernandez 6-2 6-4.
Fernandez had won 16 of her previous 17 Billie Jean King Cup matches, including all eight she played last year as she helped lead Canada to the title.
Britain are one of only four nations - alongside Australia, France and Italy - to have competed in the tournament every season since it was launched in 1963.
However, they are the only one of those teams to have never lifted the trophy, losing in four finals.
Earlier on Sunday Slovakia knocked out Australia to move into the semi-finals for the first time since 2002 after singles victories by Viktoria Hruncakova and Rebecca Sramkova gave them an unassailable 2-0 lead.
Hruncakova overcame Kimberly Birrell with a hardfought 7-5 6-7 (7-4) 6-3 win in a match lasting two hours and 30 minutes.
Hruncakova, 126 places below Birrell in the rankings, hit 32 winners while the Australian made 37 unforced errors.
World No.43 Sramkova stunned Ajla Tomljanovic with a ruthless performance, beating the former Wimbledon and US Open quarter-finalist 6-1 6-2. Tomljanovic was struggling with a knee issue.
Slovakia had shocked record 18-times champions the United States on Thursday to reach the quarter-finals.
