The Tigers will play in their first FA Cup semi-final since 1930 in the capital next weekend and will do so with their Barclays Premier League status almost certainly assured following George Boyd's first-half header.
They were, however, made to hang on in a second half when Swansea might have equalised but for a lack of cutting edge as the Tigers appeared guilty of prematurely casting their minds ahead to Wembley.
A party mood was apparent inside the KC Stadium well before the opening whistle as the home support, somewhat alarmingly armed with face masks of boss Steve Bruce, settled in for a final farewell before a rare adventure to the national stadium.
The Tigers are odds-on to return again to the capital for the FA Cup final - with League One Sheffield United lying in wait next weekend - and the feelgood factor of a memorable first season back in the top flight initially uplifted the home team before their second-half wobble.
Nonetheless this was a 10th league win of the season and lifted City to 36 points - more than the total they managed when staying up in their first top-flight campaign in 2008/09.
If survival still remains Bruce's primary objective then he will at least head to London safe in the knowledge his side have a gap of nine points over the relegation places.
With five league games remaining it therefore appears unlikely Hull will be caught - even if they still do have a tough run-in that includes Arsenal, Manchester United and Everton.
In amongst the home joy Swansea were left still with a fight to maintain their Premier League status as their poor recent record extended to just one win in their past 11 matches in all competitions.
Rookie manager Garry Monk will hardly relish the prospect of welcoming title-chasing Chelsea to south Wales next week either.
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 rearguard.
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 Liam 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 Nathan 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.