The Warriors had to wait for their final group game last season before collecting a win but they have already got themselves off the mark just two games into this year's competition after beating Exeter 20-16.
An exchange of penalties saw the Chiefs lead 9-6 at half-time but lock Swinson's try following a moment of brilliance from Niko Matawalu nine minutes into the second half set the hosts on target for victory.
Flanker Holmes added a second seven minutes from time before visiting number eight Dave Ewers ensured Exeter left with a losing bonus point that sees them draw level with Toulon at the top of Pool 2.
The hosts welcomed Exeter to Scotstoun in fine fettle. Five wins out of five had earned them top spot in the RaboDirect Pro12 while they even managed to pick up a four-try bonus point in their 51-28 defeat to reigning champions Toulon in their Heineken Cup opener.
Their attacking options were boosted by the inclusion from the kick-off of Canadian wing DTH Van der Merwe after he came off the bench to score a brace of tries in France, with Fijian international scrum-half Matawalu reverting to his favoured number nine slot having been pushed out wide recently.
But the Chiefs were themselves fresh from an eye-catching 44-29 demolition of Cardiff Blues and Glasgow boss Gregor Townsend was forced to admit before the match his side were facing one of the form teams in Europe.
And the English outfit were certainly not slow out of the blocks, capitalising on home skipper Al Kellock's infringement for not releasing with Gareth Steenson's first penalty just 90 seconds in, but the fly-half's opposite number Ruaridh Jackson soon had Glasgow level with his own successful kick.
Steenson was spot on with his next effort from wide on the right but the visitors were lucky not to lose hooker Chris Whitehead to the sin bin after his late hit on Matawalu as the Glasgow man chased his own kick, although Jackson did square things up for a second time from the resulting penalty.
Exeter's pack were exerting some real pressure on the home side and Warriors' attempt to illegally slow the ball down only allowed Steenson to continue the kicking duel with another attempt between the posts.
However, a first surge into opposition territory from Van der Merwe cost Steenson a yellow card as he was penalised by French referee Romain Poite for dangerous play at a ruck.
His 10-minute banishment cost the visitors no points, though, as Glasgow continued to waste decent openings and only put themselves under pressure with basic handling errors.
Their cause was not helped by the intermittent showers that had the ball bouncing about like a block of wet soap but Matawalu's skills at close quarter offered Glasgow hope at the break as they went in 9-6 down.
Jackson fluffed a chance to level four minutes after the restart as he miskicked an early penalty but Matawalu soon proved his worth in the half-back slot.
Glasgow's surge into the Chiefs' 22 saw the visitors backed up but well positioned.
Matawalu freed some room with an exquisite pivot before Swinson picked up and drove the ball under the posts to hand Jackson an easy kick for the extra two points.
Where the Aviva Premiership side had been dominant in the first 40 minutes, Glasgow were now bossing proceedings.
They thought they should have had a second try after 69 minutes but, after consulting TMO Herve Dube, Poite called play back after spotting a forward pass from Jackson just before wing Tommy Seymour dived over in the corner.
Their disappointment was not to last long as a wonderful break saw Sean Maitland, Van der Merwe and Jackson combine before Holmes was played in to dot down his first try for the club.
Jackson again knocked over the conversion but Glasgow nerves were set jangling almost immediately as Ewers barged over for a late Exeter try, converted by substitute kicker Henry Slade.
But there was no further drama as Glasgow held on for the win and now travel to Cardiff on December 6, while the Chiefs host Toulon a day later.
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