A Gareth Davies try plus a superb team move finished by Josh Turnbull extended their lead before Connacht hit back with tries from Fionn Carr and Robbie Henshaw to get within two points.
Gareth Maule's try settled nerves for the Scarlets but Eoin McKeon dotted down at the death to ensure Connacht did not come away empty handed.
Scarlets went on the attack from the off and got off to a dream start after three minutes with Priestland starting a move with a well-placed pass over the head of Dan Parks to free up the left wing.
Jordan Williams took up the baton before feeding in centre Davies to stride over, with Priestland adding the extras.
Parks cut the gap to four points soon after with a penalty but Scarlets were showing their intent to play their rugby.
Priestland then missed a chance to extend his side's lead when his 15th-minute penalty fell wide before Parks tried to play in Tiernan O'Hallaran in the corner but the former Scottish international's kick went into touch.
Parks was on target with his next kick, splitting the posts on 22 minutes to take his side back to within a point but Priestland replied with his own two minutes later.
And while Priestland hit the post shortly afterwards with another kick, he made no mistake with two successive three pointers to re-establish his side's seven-point lead on 34 minutes.
Connacht's best attacking move came just short of the half-time break with the ball carried out of their own 22 before Carr took hold on the wing but his chip forward ran just too long to dot down.
Scarlets began the second half in the same attacking vein as the first, and while Priestland missed a penalty on 44 minutes, Davies added his side's second try five minutes later.
In a move seemingly off the training ground, the ball was picked up from the back of the scrum and Davies fed the dummy before jinking in under the posts, Priestland with the conversion.
And it was soon three tries for Scarlets as defence was turned into attack in devastating fashion, Liam Williams doing well under pressure in his own 22, before Kristian Phillips burned his way through the centre leaving Turnbull to canter in.
Priestland missed the conversion, and Connacht replied immediately, with Marmion showing quick thinking to tap and go before squeezing over the line.
The TMO confirmed the try, although Parks also missed the conversion with the scores at 25-11 with a little over 20 left on the clock.
The visitors pushed again, with Miah Nikora sending a beautifully-weighted kick into the corner for Carr to touch down, with the former adding the extras.
And Connacht were not just settling for the losing bonus point, as after Michael Swift found room down the right with Henshaw on hand to crash over five minutes from time.
But faced with a tricky conversion on the near touchline, Nikora saw his kick drift agonizingly short before Maule calmed late Scarlets nerves with their bonus-point try and McKeon had the last word following strong work by Henshaw.
|Scarlets Score Card|
|Connacht Rugby Score Card|