Evans beat Adrian Mannarino 7-5 6-4 in a little under two hours to give Britain the lead, firing 14 winners and converting three break points as the Frenchman made 22 unforced errors.
Norrie defeated Arthur Rinderknech 6-2 7-6(8), clinching victory on his second match point in an absorbing tie-break in which the Brit fought back from 4-1 down.
“It was a difficult match especially with a lot on the line,” Norrie told Eurosport. “He came up with some great volleys on the tie-break there. The team behind me were great, especially with no fans here, it wasn’t easy.”
“It was a tough game early on and I found a lot of confidence – he went through a little patch where he missed a lot and I took advantage of that. All credit to the guy to make it to the tiebreak and I was fortunate to sneak through there.”
France hit back in the doubles, Rinderknech and Nicolas Mahut downing Joe Salisbury and Neal Skupski 6-1 6-4.
Czech Republic are next up for Britain in Sunday’s matches, being played behind closed doors due to Covid-19.
The group winners and two best runners-up qualify for the quarter-finals. Any victory will secure GB a spot in the next round, with France’s earlier win over the Czechs meaning a 2-1 defeat will lead to a set-percentage tiebreaker.
Additional reporting by Reuters.