Meslier's mistake capped a thrilling encounter between the rivals, which had promised much and did not disappoint in front of another bumper crowd of 41,769.
Sunderland have been revitalised and reinvigorated under new French head coach Regis Le Bris, while pre-season promotion favourites Leeds are bidding to bounce back after their play-off heartache against Southampton at Wembley in May.
The touchpaper was lit by 17-year-old Rigg. He had scored the winner in the Wear-Tees derby against Middlesbrough here 13 days previously and had the stadium rocking once again.
Jobe Bellingham had already shot over at the end of one good move, before another fine move resulted in the opener.
Right-back Trai Hume got to the line and fired across for left-back Dennis Cirkin arriving at the far post to force a reaction stop from Meslier, but Rigg was on hand to put the loose ball home.
Leeds had been overwhelmed by the home side's fast start, but they fashioned a terrific 22nd-minute equaliser as Willy Gnonto spotted Piroe's clever run and delivered the perfect ball which the Dutch striker headed low past Anthony Patterson for his fourth of the season.
It ended Sunderland's run of clean sheets at home as they became the 92nd and final club in the top four divisions to concede a goal at home this season.
Suddenly the visitors had the momentum and Brenden Aaronson and Firpo got in the way of each other from only six yards out when a goal seemed certain, while Largie Ramazani headed down and over from close range from another gilt-edged chance.
But Sunderland could have gone into the break ahead as Bellingham was twice close to scoring - one effort that Joe Rodon deflected just past the post and a second curling strike wide.