Former England winger Stone entered the fray shortly after 61 minutes to reinvigorate the Leeds attack.
But it was Gary Kelly who swung in a dangerous cross from the right for Healy to head home in emphatic fashion.
Crewe's best effort came shortly before the half hour mark when Gareth Taylor shot over from the edge of the box.
Leeds manager Kevin Blackwell:
"Steve Stone came on and straight away he showed the quality he's got.
"Stoney is nowhere near as fit as he needs to be but he's intelligent and he gets round the pitch so well.
"It's a good time for him to be showing us he can perform with the play-offs coming up.
"Stoney has worked his socks off in the gym and comes in every morning to the gym at nine o'clock in the morning and stays until six to seven o'clock at night.
"The biggest factor about Stoney is his professionalism and desire to get fit because it has been a career-threatening injury - and many people would have called it quits after the career he's had."
Crewe manager Dario Gradi:
"When you look at the situation that both clubs find themselves in, I thought it was a decent game for the supporters because both sides tried to play football and tried to win the game.
"I thought it was played in a good spirit and it would have been entertaining for the fans, although there were not a lot of clear-cut chances.
"We just got caught out for the goal and gave Healy too much space as the cross came in.
"Other than that I thought the back four defended very well and I was pleased with the effort the players showed.
"We gave it a real good go and maybe we were a bit unlucky not to grab a point."
Leeds: Sullivan, Kelly, Butler, Kilgallon, Crainey, Miller (Stone 61), Derry, Bakke, Lewis (Blake 66), Healy, Graham (Moore 41).
Subs Not Used: Gregan, Douglas.
Goals: Healy 74.
Crewe: Turnbull, Moss, Bougherra (McCready 81), Foster, Otsemobor, Bell (Roberts 86), Lunt, Johnson, Vaughan, Taylor (Higdon 82), Varney.
Subs Not Used: Tomlinson, Rodgers.
Ref: N Miller (Co Durham).