After heading the final practice session on Saturday, the French manufacturer's two cars both finished inside the top six within half a second of Kamui Kobayashi's polesitting Toyota GR010 HYBRID.
Jean-Eric Vergne was 0.304s shy in the #93 Peugeot that will start tomorrow's six-hour race from fourth, while Menezes was three places further back and 0.422s down in the sister #94 example.
Although it has previously started on the front row at last year's Bahrain finale, it marked comfortably the best qualifying performance of the season for Peugeot and Menezes believes the programme's efforts to widen the working window of its 9X8 is paying dividends.
Asked by Autosport after qualifying if his feeling in the car is the best it has been to date, Menezes replied: "Yeah I think we’re starting to have the feel of a proper race car.
"I know that sounds mean, but let’s say we were struggling with a lot of critical problems that were disturbing the balance.
"The car is handling quite well and it has been for a while now. In Le Mans we took the lead for a few hours which was positive.
"Obviously when the condition went to full dry we were struggling in pace. We closed that gap here, we just need to keep working with our heads down."
Menezes, who admitted to being "slightly disappointed" to be beaten by Vergne, added that the qualifying showing demonstrated Peugeot is "another step closer" to reaching its targets a year on from the 9X8 car's race debut at Monza.
"We’re not where we want to be yet, we’re just fighting to win and to be the fastest car on track more consistently," he said.
"We’re not there yet, but we’re making steps in that direction.
"I wouldn’t say it’s our best track, but our performance is improving everywhere we go and that’s a positive sign."
Speaking on Friday, his team-mate Nico Muller also expressed his belief that Peugeot had made progress on its driveability.
"We always try to get the car working in a wider range," said the Swiss.
"These cars when they are in the optimal window, I think they all can perform very well. But in endurance racing the conditions change constantly and you have to be flexible and react to those changes of conditions and make the window of your car as wide as possible.
"That's what we've been focusing on. We've tried to find drivability in the car to make sure the drivers can get in and immediately extract the maximum out of the car.
"We're still in that process but we've progressed on that quite a lot."
