The 21-year-old qualified in 10th for tomorrow's race, one better than Verstappen who did not advance to the final session.
Verstappen was pipped by just .007 seconds in the second session by Lawson who drove a best time of 1 minute and 32.268s.
Teammate Yuki Tsunoda finished in 15th.
"It's obviously exciting being in Q3, but we didn't maximise the performance, and it's something we have to look at. It's a shame for Yuki because we have a strong package, and he looks comfortable in the car,"
Lawson said he is still learning and believes there is more to come.
"For me, it was warm-up related because I didn't feel the jump in grip for the last lap and there was lots of traffic in the last sector, so that was tricky."
The Aucklander has been racing as a replacement for the injured Daniel Ricciardo who broke his hand in practice for last month's Dutch Grand Prix.
The Singapore Grand Prix begins at midnight on Monday.
"I'm starting in a good position in P10, but two hours is a long race and will be challenging. We'll focus on maximising our car and driving our race."
