Sunderland took an early lead at the Stadium of Light through teenager Chris Rigg but Leeds levelled with Joel Piroe's header.
The visitors took the lead shortly after half time through Junior Firpo's low strike.
That looked to have sealed victory until Meslier allowed Alan Browne's harmless, deflected cross to bounce past him and spin into the net in the eighth minute of stoppage time.
"I've worked in professional football for 30 years but it's the first time I've experienced that," said Leeds boss Daniel Farke.
"Normally the goalkeeper takes the ball, clears it and the referee blows the final whistle and the game is won, but not this time."
Farke added: "I thought maybe today was the day our luck changed but it's another sucker punch and you have to get on with it."
Browne admitted he felt "embarrassed" to see his speculative, aimless ball evade the Leeds goalkeeper.
"I thought the boys would be fuming (after his seemingly wasted ball into the penalty area) but the keeper chucked it in the net. I am almost embarrassed but I'll still claim it," Browne told Sky Sports.
"In the end we got the lucky break, whether we deserved it or not I am not sure. To be fair Leeds came at us. We didn't really open them up and create the chance we needed."
The result kept Sunderland on top of the table with a two-point lead over Burnley while Leeds moved into third, a point further back.
