The Briton set an impressive time of one minute 41.223 on the soft tyres, 0.222 quicker than Red Bull's four-time world champion Max Verstappen, in a tricky session with gusting winds and cars slipstreaming.
McLaren's championship leader, and last year's race winner in Baku, Oscar Piastri was third quickest but 0.254 off the pace in the hour-long session at the 6.003km City Circuit.
Piastri, winner of seven races so far this season, leads Norris by 31 points while McLaren could clinch the constructors' championship on Sunday with an unprecedented seven rounds of racing remaining.
Seven-time world champion Lewis Hamilton was fourth for Ferrari, after topping Friday's second practice, with Mercedes' Kimi Antonelli fifth and the Italian teenager's teammate George Russell sixth.
Williams' Alex Albon was seventh, ahead of Haas's Oliver Bearman, Racing Bulls' Liam Lawson and Ferrari's Charles Leclerc.
Leclerc will be chasing his fifth successive Baku pole position later on Saturday, with Ferrari and Red Bull looking like they could give McLaren a fight at the top.