Garry Monk's side were well on course for the top six heading into the international break but a return of four points from five games has seen them drop out on goal difference.
Wolves beat Brentford and Fulham on the road in March and they were full value for their half-time lead against an out-of-sorts Leeds.
Dicko's strike was only the fourth goal Leeds had conceded at Elland Road since November and the Whites went into Saturday's clash having won 10 of their previous 12 home games.
But Leeds were unable to draw on that record as they suffered their fifth home defeat of the season.
Wolves had eased their relegation fears with five straight wins but the mid-table visitors would have been determined to end their season on a high after back-to-back defeats.
Paul Lambert's side enjoyed a bright start and had the first sight of goal when George Saville - a goalscorer in last season's 2-1 loss at Elland Road - glanced a header well wide.
Leeds immediately threatened on the break but they were indebted to Robert Green for keeping out Andreas Weimann's low shot with his legs after he was left clean through following a collision between Gaetano Berardi and Pontus Jansson.
Green had to get down low to push Dave Edwards' low shot behind as Wolves continued to take the game to a Leeds side struggling with the away team's care-free approach.
Kyle Bartley threw his body in the way of a Saville strike with United again stretched before Dicko sent a 20-yard shot whistling past the right-hand post with Green beaten.
Berardi fired an ambitious volley into the crowd and Jansson headed over from a corner as Leeds got a foothold in the game.
But the hosts fell behind in the 38th minute when Dicko ran into space created by Jansson's charge upfield to latch onto Ben Marshall's through ball and roll a shot beyond Green.
Chris Wood twice threatened to get a shot off as half-time approached but the ball would not drop for the Championship's top goalscorer as Leeds' frustrations continued.
United were brighter after the break with Luke Ayling firing well over and Richard Stearman adjudged to have not deliberately handled Wood's cross.
Winger Hadi Sacko replaced defensive midfielder Liam Bridcutt in an attacking move by Monk before Jansson headed Pablo Hernandez's corner over with Leeds on the front foot.
For all United's pressure they had failed to test former goalkeeper Andy Lonergan until the 72nd minute when substitute Souleymane Doukara's header forced him to tip over.
Leeds were no closer to forcing an equaliser with full-time looming and were in desperate need of inspiration or a moment of good fortune.
Wolves substitute Kortney Hause headed Kemar Roofe's looping header off the line under intense pressure and when Wood headed over in stoppage time Leeds knew it was not their day.