The Tigers, who sat level with the Swans at kick-off, will be feeling a lot more secure now that they have clambered into 12th spot and will head to Wembley for next Sunday's FA Cup semi-final date with Sheffield United in buoyant mood.
Steve Bruce's side claimed the spoils on Saturday six minutes before half-time when Liam Rosenior worked space down the right and hung over an inviting cross for Boyd to attack at the back post and power a header past Michel Vorm.
The best opportunity for Swansea to level fell to substitute Nathan Dyer midway through the second half, but he fluffed his lines by firing wide after being picked out by a cross fed to the edge of the box by fellow winger Wayne Routledge.
With Wembley in mind Bruce exercised caution in his team selection, resting playmaker Tom Huddlestone due to a slight back strain.
Alex Bruce also missed out with Boyd and Maynor Figueroa slotting into a switched 4-4-2 formation.
Swansea showed no change from the team that beat Norwich 3-0 last weekend.
Hull's fans were in full voice from the first whistle and would have had an early goal to shout about too, but for Michel Vorm's quick reflexes.
Shane Long ran on to Figueroa's perfectly-weighted cross to the back post only for Vorm to turn his side-footed volley wide.
Hull had most of the initial possession but Swansea were a threat on the break and Jonathan De Guzman should have at least tested home goalkeeper Steve Harper when released on goal.
The midfielder ran 30 yards unchecked into the area before cutting inside his marker, but delayed his shot and allowed Hull's back-tracking defence to block.
Wayne Routledge then failed to get enough elevation on a lobbed shot when he got to a through-ball before Harper.
Hull's attacks were more sustained, although they were forced to the flanks to try and break down Swansea's rear guard.
Curtis Davies headed an Ahmed Elmohamady cross wide while Long lost a Boyd centre in the sun when in space at the back post.
Hull's perseverance paid off after 39 minutes, thanks to a lovely Rosenior cross from the right.
His delivery to the back post was set-up for Boyd to jump over a static Angel Rangel and the Hull midfielder did so easily before directing his header back across Vorm.
Swansea were brighter after half-time and Hull needed to be awake to cut out deft passes in behind first from Routledge and then Wilfried Bony.
When Swansea turned to the flanks Davies won everything in the air.
The away side's increasing grip on the game forced Bruce to reinforce midfield with Sone Aluko dropping deeper after coming on for Long just after the hour.
Still Swansea threatened and Jonjo Shelvey shot straight at Harper, who momentarily spilled the low shot with Routledge lurking.
Routledge then outmuscled Figueroa and played a pass across for Dyer who screwed his shot horribly when he had plenty of time.
Hull's attacks were at a premium so when Jelavic blazed a free-kick high over the groans from home support were audible.
Bony stabbed over when Shelvey played him behind the defence and then was almost given a tap-in by some hap-hazard home defending only for Harper to recover the loose ball.
It was hardly a convincing way to sign off before their FA Cup venture, but the Tigers at least held on to head to London in a winning mood.