The under-fire England winger smashed home in the second half before the skipper made it two with a curling effort to book a place in the last four of the Capital One Cup
Young has lost his way at Old Trafford, but he produced a moment of quality in the second half on Wednesday to put United in front and end a goal drought that stretched to May 2012.
Evra made sure of the win with a lovely right-foot finish to enhance David Moyes' chances of winning a trophy in his first season at United.
Miserable conditions at the Britannia Stadium led to referee Mark Clattenburg taking the players off the pitch for just under 10 minutes in the first half.
Stoke beat United en route to winning this competition in 1972, but its wait for a second major trophy continues and the Staffordshire club has now lost 12 of its last 13 games against United.
United was without Wayne Rooney due to injury but Moyes is hopeful he will return in the near future. Adnan Januzaj and Ryan Giggs were left out as Chris Smalling, Anderson and Young started.
Thomas Sorenson, Erik Pieters, Jonathan Walters and Wilson Palacios came into the Stoke starting lineup.
Strong winds and driving rain ensured it would not be an evening for the faint hearted. Stoke was almost caught cold after only two minutes, but Young could only find the side-netting with a right-foot strike from the right side of the penalty area.
The weather deteriorated so badly that Clattenburg halted the game after 28 minutes amid a hailstorm and swirling winds.
Clattenburg led the players back out just under 10 minutes later, but the quality of football was no better as both sides struggled to retain possession.
Geoff Cameron produced a rare moment of quality just before halftime when he swung in a dangerous cross from the right, but the ball skimmed off the top of Walters' head and away to safety.
Moyes replaced Anderson with Javier Hernandez 11 minutes into the second half with no sign of a breakthrough.
And Hernandez was involved in the opening goal six minutes after he came on as Young played the ball into him and his touch fell nicely into the path of the onrushing winger, who lashed a vicious right-foot drive beyond Sorenson from just outside the area.
Young was brimming with confidence and the former Watford and Aston Villa man embarked on a run towards the area before getting his finish all wrong.
Pieters whipped an inviting ball in but Walters somehow failed to get a touch before the Republic of Ireland striker's long-range strike was tipped over by David de Gea.
And Evra ended any hopes of a Stoke comeback 13 minutes from time when he burst into the box and curled a measured right-foot finish into the far corner.
Stoke captain Ryan Shawcross limped off injured to complete a miserable night for the home side.