The Scarlets had led 32-18 with 17 minutes left on the clock, but spearheaded by second-half replacements Trimble and powerhouse number eight Nick Williams, the Ulstermen came surging back to share the spoils.
It was a fantastic advert for the Pro12 with both sides willing to give the ball air, running in four tries each to finish with three points apiece for their efforts.
A breathless opening saw three tries scored inside the first 14 minutes.
The opener came after just two minutes and was scored by wing Harry Robinson - one of three new signings making their Scarlets debut.
A clever chip and chase by Wales centre Scott Williams set up the attack, but after he spilled the ball, Ulster attempted to counter from deep.
The Irish province were then turned over and with men over, slick Scarlets handling put the former Blues flyer over in the corner and fly-half Rhys Priestland converted.
Ulster's response was impressive. From an attacking line-out, the white forwards charged at the home line and the Scarlets defence was eventually stretched enough for South African debutant Franco van der Merwe to drive over for a score converted by fly-half Ian Humphreys.
Priestland landed a penalty straight from the restart, but back came Ulster with a second try inside four minutes - from Van der Merwe's second-row partner Dan Tuohy. It was a score created by a sublime break from centre Stuart Olding, who proved a handful all afternoon.
But in keeping with an electric first half, that only prompted the Scarlets into action.
With Scotland flanker John Barclay to the fore, the home side piled on the pressure and eventually Ulster's defence succumbed with number eight Rory Pitman - a summer arrival from London Wasps - touching down out wide.
Priestland was again on target with the extras before Ulster were denied their third try.
South African full-back Louis Ludik - another on debut - thought he had crossed, but the Scarlets had felt there had been obstruction in the build-up, a view referee Peter Fitzgibbon eventually agreed with after a lengthy TMO review on the big screen.
Humphreys reduced the deficit to 17-15 with a long-range penalty, but it was the Scarlets who had the final word of a superb half of rugby.
With Ulster on the attack, lock Jake Ball came up with the ball, fed scrum-half Gareth Davies and last year's top try-scorer raced 60 metres to the line with Priestland's conversion making it 24-15 at the break.
A rash challenge from Tuohy on Scott Williams five minutes after the restart saw Priestland add three more points to the scoreboard with the Ireland lock sin-binned for the offence.
However, Scarlets also found themselves a man short straight from the restart when full-back Liam Williams was shown yellow for a deliberate knock down as Ulster pressed and Humphreys reduced the deficit to 27-18.
A great counter from replacement Gareth Owen saw Davies claim his second on 62 minutes, but to their credit Ulster refused to throw in the towel.
Ludik crossed wide out on 68 minutes and Humphreys converted to narrow the gap to just seven points.
Then, after Liam Williams was dismissed a minute from time , Trimble was put away and Humphreys made no mistake with the conversion to rescue a share of the points for Ulster.