The centre-back, who signed from Tottenham last summer, rose unmarked and onside on 37 minutes to head home Ahmed Elmohamady’s cross.
Liverpool struggled to create clear-cut chances, going closest in the first half through Philippe Coutinho and Jordan Henderson on a night Steven Gerrard was rested.
The home side worked tirelessly to keep the Champions League hopefuls, now short of the top four by seven points, at bay and a second successive Premier League victory sees Hull move four points clear of safety with four games remaining.
Hull boss Steve Bruce fielded an unchanged side from the one that beat Crystal Palace at Selhurst Park on Saturday and they continued the huge degree of commitment and work-rate that bore that victory.
Liverpool manager Brendan Rodgers started Mario Balotelli for the second successive time, but he was ineffective and frustrated as early as the first minute, clashing with Tom Huddlestone and then Jake Livermore.
Hull striker Dame N’Doye threatened first,heading a Sone Aluko cross straight at Simon Mignolet on seven minutes before Coutinho stung the hands of Steve Harper at the other end with a first-time shot from a corner.
But it was Balotelli who was arguably at fault for Dawson’s goal – playing the defender onside in his own penalty area despite three other Hull players standing in an offside position.
Henderson forced Harper into another good save low to his right when latched on to a cross from Jordon Ibe, who was lively before he was replaced in the second half.
And Balotelli was further exasperated on the stroke of half-time when he demanded the ball early from Glen Johnson, but did not get it, and was then too late to latch on to the right-back’s flash across the goal-line.
The second half was introduced with a moment of comedy, as N’Doye took to the pitch wearing the shirt of Abel Hernandez and had to be alerted to the fact by referee Lee Probert, before dashing to the sidelines to switch.
But the fact Liverpool did not get the ball in the attacking third again until just before the hour-mark was no laughing matter, and they were limited to wayward shots off-target from outside the box for the rest of the game.
The Reds remain seven points short of fourth-placed Manchester United, with their Champions League hopes seemingly dissipated, but Hull lift themselves above Aston Villa into 15th in the table.
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