World number three Federer defeated Norrie 61 6-1 before teammate Belinda Bencic be at Katie Boulter 6-2 7-6 (7-0) from the women’s singles to secure the tie.
Norrie, 23, and Boulter, 22, play Serena Williams’ USA on Thursday.
They defeated Greece within their opening tie in the championship on Saturday.
The Hopman Cup comprises two singles games and a mixed-doubles competition between states within an round robin format with two sets of four.
The champions of both groups will contest Saturday’s final.