The result put eighth-placed Villa on 39 points from 24 games, six behind West Ham United in fourth and with two matches in hand. Leeds stayed 10th on 35 points from 26 games.
Leeds had 70% possession but lacked a cutting edge as most of the chances fell to Villa, who blocked the supply routes and looked the more likely to score again after Dutchman El Ghazi netted in the fifth minute.
Despite a heavy first touch after Ollie Watkins whipped in a cross from the edge of the area, El Ghazi was still able to slide the ball past Leeds goalkeeper Illan Meslier as he beat the offside trap.
The first half produced end-to-end action and El Ghazi missed three more chances with Meslier denying him twice before he shot wide from a good position, while Tyler Roberts drew a superb reflex save from Emiliano Martinez at the other end.
The pace dropped in the second half on a difficult pitch and the match became littered with fouls as Leeds midfielder Roberts and Villa defender Matt Targett got booked after an altercation.
Visiting manager Dean Smith praised his side for a gritty performance which allowed Villa to bounce back from Sunday's 2-1 home defeat by Leicester City.
"We made a really good start and got on the front foot early," he told Sky Sports.
"Our intensity levels were certainly a lot better than they were last week. In the first half against Leicester there wasn't as much intensity as I'd like. Today was a different game.
"We restricted them to very few chances in the second half. We wanted to play with safety first and we did that."
Leeds boss Marcelo Bielsa rued his team's toothless performance up front.
"The crosses were not precise and the final passes weren't precise enough either," he said. "It was a second half with sustained dominance throughout, more than the first half, [but] it couldn't be translated into a goal."