Williams’ Carlos Sainz was fifth, while Racing Bulls’ Isack Hadjar took seventh with his final lap of the session. Mercedes’ George Russell was ninth, ahead of team-mate Kimi Antonelli and Ferrari’s Lewis Hamilton.
Norris took to the escape road at Turn 14 at the end of Las Vegas’ famous Strip twice on consecutive laps as he fought to find pace in his McLaren.
He also hit the wall early in the session, although he escaped without damaging the car.
Piastri, 24 points behind the Briton at the head of the championship, had his off-track moment at Turns Eight and Nine around the Sphere.
The Australian was 0.192secs slower than Norris, continuing the performance trend between the two in most recent races.
Norris has gained 58 points on Piastri in the past six races, to take control of the season and put himself within sniffing distance of the championship.
Leclerc was quick all session, but Hamilton ended up 0.759secs behind his team-mate.
The session is unlikely to be representative of the rest of the weekend as the track is so dirty and will improve as the sessions tick by.
Second practice is at 04:00 GMT.