Lawson has quickly built a Formula One rivalry with Red Bull senior driver Sergio Perez, who qualified 16th, the latest in a string of disappointing performances from the struggling Mexican.
Yuki Tsunoda joined Perez in the bottom five and will start 17th.
Lawson and Japanese driver Tsunoda are challenging Perez for his seat in the top Red Bull line-up.
McLaren's Lando Norris took pole position for Sunday's Qatar Grand Prix sprint, with Mercedes' George Russell denying the Formula One leaders a front row lockout by pushing Oscar Piastri down to third.
McLaren is leading Ferrari in the championship by 24 points with two rounds remaining and a first constructor's title since 1998 tantalisingly close.
Ferrari's Carlos Sainz qualified fourth with team-mate Charles Leclerc fifth and world champion Max Verstappen sixth for Red Bull and Lewis Hamilton driving for Mercedes seventh.
Lawson's final attempt was invalidated for exceeding track limits.
The Kiwi scraped into Q3 with a buffer of just 0.004 seconds over Fernando Alonso in Q2 and survived Q1 by a mere 0.013 seconds at the expense of Perez.
Alonso was knocked out of Q2 in 11th, joining Alex Albon, Valtteri Bottas, Lance Stroll and Kevin Magnussen, respectively, as the eliminated drivers.
In Q1, Lawson and Albon set identical times, which relegated Perez to yet another early exit from qualifying.
The Qatar Grand Prix Sprint Race begins at 3am (NZ time) on Sunday, before Grand Prix qualifying at 7am.
