Both teams went into the clash needing three points to close the gap on sixth-placed Leeds, however it was an early strike from Delfouneso that separated the pair.
The 20-year-old, signed in the build-up to the encounter, poked home from close range to seal a win that keeps Eddie Howe's team well poised to make the top six.
Hull launched a second-half riposte but found Lee Grant impassable as he ensured a damaging defeat was inflicted on Nigel Pearson's promotion hopes.
The Tigers made two changes from the team that beat Nottingham Forest at the weekend. Nick Barmby came in for Jay Simpson, while goalkeeper Brad Guzan returned on loan from Aston Villa to replace the injured Vito Mannone.
Villa team-mate Delfouneso added to the potency Howe's front line needed to keep abreast of the division's front-runners.
That meant Chris Iwelumo dropped to the bench as Burnley began with purpose and seized the upper hand thanks to a fifth-minute opener straight off the training ground.
A corner was driven low into the box and although Jay Rodriguez's low drive was blocked, the ball fell right at the feet of Delfouneso, who prodded in from two yards.
It took 15 minutes for the hosts to offer a response and even then it was a speculative hit from James Harper that was always going over.
Burnley continued to look at ease going forward and their free-flowing, confident play had them in comfortable command - much to the ire of the Hull crowd.
Anthony Gerrard had a sniff of a chance when his head grazed an Andy Dawson cross but Clarets keeper Grant had little more than handling practice to contend with.
Counterpart Guzan did well to clutch an awkward shot from Ross Wallace that spat up and nearly bounced past the American as Burnley looked good value for their slender lead.
Howe will have been disappointed his charges failed to put the sluggish Tigers to the sword, given they had to rely on Michael Duff's terrific goal-line clearance to keep their advantage intact after Matty Fryatt had gone clean through 10 minutes before the break.
An Aaron Maclean header failed to find the target and Corry Evans volleyed narrowly wide but could not test Grant after the restart as Hull looked to mount a fightback.
But they failed to weave any meaningful possession together - in contrast to their recent displays - and their best hope of a leveller came when Dawson steamed in to blast a volley goalward, only to be thwarted by Grant's excellent save.
Former Hull midfielder Dean Marney almost added to the indignation by shaving the post with a driven shot from outside the area.
Burnley soaked up the pressure and stood firm in face of City seeing the majority of the ball. When cracks did appear, Pearson's team failed to make the most of them.
Evans was played in and with the goal at his mercy, saw Grant come to the rescue again and stick out a hand to keep Burnley ahead.
David Amoo, loaned from Liverpool, was introduced to inject pace but the visitors had the next opening as substitute Wade Elliott cleared the crossbar.
Burnley absorbed further Hull attacks while looking threatening on the break, but it was the contribution of Delfouneso that proved decisive in this play-off tussle.