Joe Gelhardt stepped off the bench to score a late winner for Leeds and boost the Whites' hopes of avoiding relegation from the Premier League.
After Gelhardt restored United's lead, Meslier's opposite number, Tim Krul, joined the Norwich attack for a corner at the other end of the field in the game's dying minutes.
The Yellows 'keeper volleyed the ball in behind the Whites' back line where it fell to Norwich's top scorer Teemu Pukki, but Meslier was on hand to stop the striker's effort.
The save kept the scoreline at 2-1 and ensured Leeds claimed all three points from a crucial bottom-of-the-table battle.
The win, sealed in front of a raucous crowd of 36,321, puts United four points clear of the drop zone and prevents bottom-placed Norwich from gaining ground on them.
At the final whistle, the joy in the stands was far removed from the dismay which followed the Whites' lacklustre performance against Aston Villa on Thursday night.
For the good feeling, Dallas said, the Elland Road faithful have to thank not just the heroics of Gelhardt, but also the Whites' 22-year-old goalkeeper.
"It's one more of relief, I think," Dallas said.
"Even when we've scored to go 2-1 up... we've got to credit Illan, again, at the end.
"He's come under criticism for goals we've conceded this year and he's been a bit hard on himself because obviously he's the goalkeeper
"But he saved that one with his head the end - that could've been 2-2 and it could've been a different feeling."