The home side raced into a 2-0 lead after half an hour, only for Bute skipper Hector Whitelaw to haul his team level with two goals in the space of just four minutes.
But Neil Campbell edged Inveraray back into the lead on the stroke of half time,
and the hosts moved up a gear in the second period to book their place in the final of a competition they've won on 20 occasions in the past.
Inveraray's indifferent form going into the game - the team had picked up only three points from seven Premier League fixtures in the opening weeks of the season - suggested that Bute might never have a better chance to reach the final of a competition they themselves have never won.
But Inveraray gave notice of their intentions within 60 seconds of the throw-up on a glorious afternoon for shinty, Garry MacPherson seeing his shot deflected high over Bute keeper Kevin Queen's crossbar for the first corner of the match.
The Argyll team enjoyed most of the early pressure and possession, and it took Bute until the tenth minute of the match to register their first serious shot on goal, Robert Walker watching as his fierce drive was beaten away by Inveraray keeper John MacKenzie, Robert Ferguson flicking the rebound just wide of the target.
David Whitelaw sent an ambitious effort from wide on the right just over the bar as Bute continued to work their way back into the match, but just two minutes later the islanders' defence was caught out by a long ball over the top from Inveraray skipper Douglas Dando, and MacPherson had all the time and space he needed to thump the ball into the net.
Queen did well to palm the ball away from a corner after David Robertson embarked on a battering-ram run towards the Bute goal from wide on the left, but the hosts' second goal wasn't too long in coming, Campbell snapping up the rebound after Queen parried a MacPherson shot.
A third Inveraray goal would surely have killed the match as a contest - but instead it was Bute who gave themselves a golden opportunity within a minute of the restart after Whitelaw was hauled down inside the enlarged penalty area, one of several rule changes applying only to the Celtic Cup tournament.
There was a delay of several minutes before the penalty could be taken, Inveraray full-back Scott Robertson requiring treatment for an injury sustained in the challenge which brought down Whitelaw, but the wait didn't upset the Bute captain, who made no mistake from the spot.
And just three minutes later the travelling Bute support were given even greater cause for celebration when David Whitelaw saw his shot deflected wide for a corner - and from the short set-piece, Hector Whitelaw held off his marker and won himself just enough room to fire a spectacular shot into the far corner of the net from an incredibly tight angle.
Walker saw his well-hit shot whistle just inches wide of the post after an intelligent lay-off by David Whitelaw as Bute tried to complete their comeback - but instead it was Inveraray who grabbed the crucial half-time advantage, Campbell hammering a loose ball into the net after an untidy scramble in the Bute penalty area.
Inveraray made their intentions abundantly clear when they went on the attack straight from the throw-up at the start of the second half, and only a superb block by Queen prevented MacPherson scoring a fourth for the hosts barely a minute after the restart.
Robert Ferguson watched in agony as his fierce drive took a wicked deflection off an Inveraray defender on 54 minutes, MacKenzie scrambling back to bundle the ball out for a corner, but that Bute attack would prove to be very much an isolated incident as Inveraray took a stranglehold on the game from that moment on.
Even so, Bute knew they would remain in the game for as long as the gap between the sides stayed at a single goal, though on more than one occasion it took a last-gasp intervention from the visitors to prevent the loss of a fourth goal.
No intervention was more last-moment than Queen's full length save to deny MacPherson from the penalty spot after James Craig was adjudged to have brought down David Robertson inside the box on 58 minutes - though a remarkable passage of play four minutes later, which saw three Inveraray shots on goal in the space of no more than 15 seconds all blocked by frantic Bute defensive lunges, ran it close.
Robertson was denied by another fine save from Queen as the second period reached the halfway mark, but it was increasingly looking like a question of when, rather than if, Inveraray would put the tie out of their visitors' reach - and that moment duly arrived on 75 minutes when a nice passing move saw the ball set up nicely for substitute Grant Griffin to slip it into the net from six yards' range.
By this time Bute were rapidly tiring in the remorseless heat, and any lingering doubts about the outcome of the game looked to have been well and truly dispelled three minutes later when Davie MacPherson was allowed far too much room 30 yards from goal, and he gave Queen no chance with a sweetly struck drive into the top corner of the net.
Bute, though, refused to crumble in the face of the Inveraray onslaught, and straight from the restart Robert Walker's cutback was swept into the net by David Whitelaw to reduce the deficit to two goals.
A fourth Bute goal in the final ten minutes would have set up a real grandstand finish to the game, and the islanders couldn't be accused of not going for it in the closing stages, with Ferguson, David Whitelaw and substitute Ali Carmichael all coming close from distance in the last few minutes.
But the fairytale comeback wasn't to be, and referee Ronnie Campbell's final whistle after four minutes of injury time confirmed Inveraray's place in the final at Old Anniesland in Glasgow on Saturday, June 27, when the Argyll side will face Oban Camanachd.
Bute will have the chance to exact revenge on Inveraray next Saturday, June 6, when the two teams lock horns in Rothesay in the quarter-finals of the Scottish Hydro Electric Camanachd Cup, with a 2.30pm throw-up at the Meadows.
* Bute's second team had a more fruitful afternoon on the road on Saturday than their senior counterparts thanks to a 6-0 win at Ballachulish in South Division Two. Full report on that game to follow.
