In one of the brightest days of what has been a challenging season for Lawson, he improved from 17th-fastest in the opening session to be fifth-quickest around the famed Monte Carlo street circuit.
His best lap time of one minute 11.823 seconds was less than half a second slower than Frenchman Charles Leclerc, who topped the timesheets in both practice sessions for Ferrari.
It was an encouraging outcome for 23-year-old Lawson - who is still chasing his first points this year - heading into the final practice session on Saturday night (NZT), followed by qualifying on Sunday morning.
"It's been a positive day for us, it's one of those tracks where you're building up the whole weekend," he said.
"We made a good step from this morning but naturally tomorrow everyone will keep progressing and we need to keep chasing. We made a lot of changes today and we will keep working.
"I've raced here before in other Formula's but it's much better in an F1 car, it's faster and way more compliant. It's a nice car and really fun, it's special to drive around here.
"Tomorrow is the important day. We will focus on making another step in FP3 and try put ourselves in a good place for qualifying."
Leclerc fastest
Leclerc thrilled the Ferrari fans by setting the pace in his home event after overcoming an early collision and despite low expectations.
McLaren's Formula One championship leader Oscar Piastri went into the tyre wall nose-first at Sainte Devote but still ended the day second fastest, 0.038 slower than last year's winner.
The Australian split the Ferrari drivers in practice two, with seven times world champion Lewis Hamilton third.
Leclerc, who has been on pole three times in the last four Monaco races, lapped the tight and twisty street circuit with a best time of one minute 11.355 seconds in the late afternoon sunshine.
He had set the pace in opening practice with a lap of 1:11.964 - a surprise after neither Ferrari qualified in the top 10 at Imola last weekend.
Piastri, who leads Norris by 13 points after seven of 24 races, said it had been a tricky day: "When we get everything together, the pace is quite good", the Australian said.
"Ferrari look a lot stronger than they have done. But today was more on my side being quite messy."
Aston Martin's Lance Stroll collected a one-place grid drop for Sunday's race after stewards ruled the Canadian was to blame for the collision with Leclerc running into him at the hairpin.
The session was stopped for debris to be cleared with the Aston Martin requiring repairs to the rear suspension and a gearbox change.
Stewards ruled Stroll had cut across Leclerc's path, with the Canadian saying he had not heard the team's warning over the radio.
Leclerc broke his car's front wing in the incident but returned after a brief red flag period to go 0.163 quicker than Red Bull's four-times world champion and Imola winner Max Verstappen.
"Ferrari have taken a big step forward here," Verstappen said.
McLaren's Lando Norris was third and fourth in the respective sessions.
Racing Bulls driver Isack Hadjar clipped the barrier at the tunnel exit and triggered red flags with his car limping back on three tyres and a puncture.
The rookie then hit the wall again at the first corner, damaging his car's rear suspension.
"I am so stupid. I think I'm just dumb," he said.
Hamilton was only ninth in practice one after going airborne over the kerbs at the swimming pool exit while Verstappen dropped to 10th in the second session and complained about traffic hazards.
"Monaco is always quite tricky with traffic but two times today it was quite dangerous," he said.
"It could have been quite a big crash if you don't back out of it quickly or read the situation but luckily everything went well."
Mercedes' George Russell was sixth and 12th respectively.
Aston Martin's double world champion Fernando Alonso, yet to score a point this season, completed the top 10 in practice one and was seventh in the second session.
- RNZ/Reuters
