The SF-24 car was launched at the Italian team's Fiorano test track with both drivers hoping it would be easier to drive and develop through the longest of campaigns with a record 24 races.
Leclerc recalled how last year he had not been happy with the overly sensitive car's behaviour after three or four laps of the shakedown.
"This year the car feels healthier and in a better place," he told reporters after three laps "with cameras stuck all around the car and not really pushing to the limit.
"In terms of actual feeling of the very first laps, I would say I’ve had a better feeling this year than I did last year," he added, while cautioning that it meant nothing in terms of performance if rivals had made bigger steps.
The sport's most successful team, and the only one to have competed in every world championship since 1950, dropped to third overall last year behind Mercedes and Red Bull.
Leclerc said the aim for 2024 was "to be front runners all the time".
Spaniard Sainz, who is in his final year at Maranello before making way for Mercedes's seven-times world champion Lewis Hamilton in 2025, did two laps.
"It was a good installation lap, nothing happened," he said. "So far everything is going smoothly."
Sainz was the only driver last season to beat Red Bull, his victory in Singapore denying the champions a sweep of all 22 races.
The car is the first fully conceived under team boss Fred Vasseur, who took over from Mattia Binotto in January 2023, and the Frenchman said the team needed to continue last season's strong finish.
"We have also to be focused on the weaknesses of 2023 and to be much more opportunistic in terms of points scored and we will be focused on this," he added.
Ferrari ended last season with Leclerc runner-up to Red Bull's Max Verstappen in Abu Dhabi but still 17.8 seconds behind the triple world champion whose team had long given up developing their car.
Chassis technical director Enrico Cardile said every area of the Ferrari had been redesigned.
"We have taken on board what the drivers told us and turned those ideas into engineering reality, with the aim of giving them a car that’s easier to drive and therefore easier to get the most out of and push it to its limits," he said.
The mostly red car, the 70th that Ferrari have built to compete in Formula One, features yellow longitudinal stripes for the first time since 1968, along with some white.
The season starts in Bahrain on March 2.
