Palace midfielder Danns fired home through a melee of players from the edge of the penalty area in the second minute and although the Londoners were unable to build on their advantage, the onus was on Leeds and it took Simon Grayson's side until the final 20 minutes before they rose to the challenge.
Leeds substitutes Sanchez Watt and Davide Somma both went close to snatching a point for the visitors, but it was too little too late and Palace, who had Danns sent off for two yellow cards in stoppage time, held on for victory.
Considering so much was hingeing on the result at Selhurst Park the tepid nature of the contest was baffling, but Palace were happy to merely contain their out-of-sorts visitors, who have secured only two points from their last five games and slipped to ninth place, three points adrift of the final play-off spot with two games remaining.
Palace escaped relegation on the final day of last season when a point from a 2-2 draw at Sheffield Wednesday was enough to send the Owls down at their expense, and they were never properly troubled by the Yorkshiremen.
Victory lifted Dougie Freedman's side up to 47 points, six better than nearest rivals Sheffield United with Palace holding a far superior goal difference.
Palace made a flying start and took the lead courtesy of Danns' thunderbolt in only the second minute, the Palace midfielder latching on to the ball on the edge of the penalty area and smashing a shot into the bottom corner through a crowd of players.
Leeds struggled to get going and although they enjoyed an equal share of possession they failed to trouble Julian Speroni in the Palace goal until the closing stages of the first half.
Jake Livermore, on loan from Tottenham, fired off target from 25 yards, Bradley Johnson's effort from similar distance was blocked by Danns and the Leeds midfielder headed Robert Snodgrass' corner over the crossbar.
Jermaine Easter headed Palace's next best chance over the crossbar following Nathaniel Clyne's cross, while Kasper Schmeichel comfortably saved from Kagisho Dikgacoi's shot from the edge of the penalty area on the stroke of half-time.
Leeds manager Simon Grayson replaced Livermore and defender Paul Connolly with Neil Kilkenny and Watt at the break and the visitors continued to push forward at the start of the second half, but failed to get any shots on target.
Johnson drilled in an angled drive from 20 yards in the 55th minute, but Speroni had it covered, while Easter lashed a 25-yard shot over the crossbar and Dikgacoi headed Danns' corner weakly wide at the other end as the game continued to flounder.
Leeds winger Max Gradel then raced clear but fired his shot too early from the edge of the area, and Speroni took an easy catch.
Watt saw his 20-yard shot blocked by David Wright and then Speroni saved well from the striker's header following Snodgrass' cross.
He also kept out Somma's long-range effort as Leeds staged a late rally, but despite the dismissal of Danns for a second bookable offence deep in time added on, it was not enough to rescue a point.