Eintracht were reduced to 10 men in the 34th minute when Chris was shown a red card for elbowing Heiko Westermann, and the Royal Blues made their numerical superiority count when Patrick Ochs put through his own net seven minutes later.
Schalke were unable to capitalise further but held on for the victory, which gives them 11 points - one more than Hamburg, who play at Wolfsburg on Sunday.
Eintracht remain on two points from four games to lie in 17th place.
Schalke came into the match looking to bounce back from blowing a 3-0 lead to draw 3-3 in their last Bundesliga encounter against local rivals Borussia Dortmund last weekend.
Fred Rutten's team dominated the run of play early and nearly grabbed the lead in the 14th minute. Jefferson Farfan curled a ball over the visitors' wall but it blasted the crossbar.
From the hosts' next corner, Kevin Kuranyi missed from just two yards out with his heel.
Schalke pushed forward more and more and Eintracht went a man down 11 minutes before half-time when Chris' elbow on Westermann was punished.
And it took only seven minutes for the extra man to pay dividends as Farfan pounced on a mistake in the Eintracht defence, ran onto the loose ball and forced Ochs to deflect the ball into his own net.
Jermaine Jones nearly doubled the lead a minute after the break but fired narrowly wide.
Schalke were unable to mount any serious attacks for the rest of the match but had already done enough to secure three points.