Eager to pull away from the bottom three, the Tigers started the Yorkshire derby on the front foot and almost edged their noses in front in the seventh minute when Evandro's drive was blocked by Blades goalkeeper Jamal Blackman, and Dicko sent the rebound into the side-netting.
United had the chance to break into the top six with a victory and they created a few early opportunities of their own, the first of which came in the 11th minute when Ryan Leonard sliced a shot wide from 25 yards, failing to spot teammate George Baldock in a better position in front of goal.
The closest either side came to breaking the deadlock in the first period was in the 13th minute as Hull's Sebastian Larsson struck a free kick from 30 yards out, his effort bouncing less than an inch wide of the post.
There was more action in the stands than on the pitch in the remainder of the first period as a protest from the home support ground the match to a brief halt in the 20th minute.
A number of fans hurled foam balls onto the pitch to express their discontent towards owners Assem and Ehab Allam, and in the 30th minute, they blew whistles in an attempt to cause further disruption.
Billy Sharp was a live wire for the visitors early in the second period, putting the Tigers' defence under constant pressure, and coming closest to getting his name on the scoresheet in the 52nd minute as he gave his marker the slip in the box and tested Allan McGregor from close range.
Hull responded well and took the lead in the 52nd minute following a corner which the Blades struggled to clear, allowing Jackson Irvine to tee up Dicko in the penalty area, and the Mali international fired into the roof of the net from eight yards out.
Hull City have dropped 22 points from winning positions this season, but that was not to be the case against promotion-chasing United, who found clear-cut chances hard to come by in the closing stages.
The win leaves the Tigers in 20th place in the Championship going into the weekend fixtures, while the Blades remain in seventh.
HULL CITY (4-2-3-1): McGregor; Aina, Clark, Dawson, MacDonald; Larsson, Irvine, Bowen, Diomande (Grosicki 64'), Evandro (Stewart 78'); Dicko (Campbell 73')
SHEFFIELD UNITED (3-5-2): Blackman; Basham (Donaldson 81'), Stearman, O'Connell; Baldock, Fleck, Evans (Lundstram 74'), Leonard (Duffy 63'), Stevens; Clarke, Sharp