After a hard-to-read session as teams toyed with setups and upgrades Norris edged Max Verstappen in the three-time world champion's Red Bull by 0.024 seconds.
Spaniard Carlos Sainz, appearing in front of his home crowds for the last time as a Ferrari driver, slotted in third as his teammate Charles Leclerc finished outside the top 10.
Mercedes' George Russell, on pole last time out in Canada, took fourth from Sergio Perez in the second Red Bull, with the Mexican facing a three-grid place penalty in Sunday's race.
Oscar Piastri in the second McLaren, Russell's team-mate Lewis Hamilton, Alpine's Esteban Ocon, Fernando Alonso (Aston Martin) and the Williams of Alex Albon rounded out the top 10.
In hot sunshine, a welcome change after Thursday's scirocco rain and dust showers from the Sahara, Russell made an early statement of intent at a track where Mercedes celebrated a double podium last year.
Verstappen then went top by around a tenth of a second from Russell as Ferrari's Leclerc complained "the car is horrendous for now".
Oliver Bearman, taking over Nico Hulkenberg's Haas for opening practice, had a steady time of it, assembling data and taking care of a car only lent to him as he takes another step towards a possible entry in F1 next season.
The teenaged Italian-based Briton made a sensational debut when acting as super-sub for appendicitis-hit Carlos Sainz in Saudi Arabia, finishing seventh, and ended up 19th on Friday's run-out.
Midway through the session some track debris momentarily triggered a red flag just as Norris leapt to the top of the time sheets in his McLaren.
The loose front wing of Fernando Alonso's Aston Martin duly removed from the circuit, opening practice resumed.
Pierre Gasly then limped back to the pits, the Frenchman reporting "there's something wrong with the car".
A comment that could sum up the overall Alpine team's season on the day they appointed former Renault and Benetton boss Flavio Briatore in an executive advisory role - a decision met with raised eyebrows from some in the sport given Briatore's colourful history and ban over 'crashgate' at the 2008 Singapore Grand Prix.
Verstappen approaches the 10th round of the 24-race world championship leading by 56 points from Leclerc, with Norris hovering only eight points back in third.
After missing out in Monaco to Leclerc and Miami to Norris Verstappen reasserted control with a stylish win in difficult circumstances in Canada last time out and will be eager to confirm that domination on Sunday in a race where he gained his maiden F1 win back in 2016.
