The Scarlets led 32-18 with 17 minutes left but, spearheaded by second-half replacements Trimble and powerhouse number eight Nick Williams, the Ulstermen came storming back.
The first try came after just two minutes and was scored by wing Harry Robinson — one of three new signings making their Scarlets debut — with fly-half Rhys Priestland adding the extras.
Ulster’s response was impressive. From an attacking line-out, their 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.
But 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.
Humphreys reduced the deficit to 17-15 with a long-range penalty, but Gareth Davies raced 60 metres to dot down before Priestland converted, making it 24-15 at the break.
Priestland and Humphreys traded penalties for 27-18 before Gareth Davies appeared to put the game out of Ulster’s reach with his second try.
But late tries from Ludik and Trimble, as well as a late Humphreys conversion rescued a draw.
SCARLETS: L Williams, Robinson, King, S Williams, Tagicakibau, Priestland, G Davies, John, Owens, R Jones, Ball, Synman, McCusker, Barclay, Pitman.
Replacements: Owen for Tagicakibau (58), R Williams for G Davies (65), R Evans for John (61), P Edwards for R Jones (73), A Shingler for McCusker (52).
ULSTER: Ludik, Allen, Payne, Olding, Gilroy, Humphreys, P Marshall, Black, Herring, Herbst, Tuohy, van der Merwe, Diack, Henry, Wilson.
Replacements: Trimble for Allen (51), L Marshall for Payne (49), Heaney for P Marshall (79), Warwick for Black (41), Fitzpatrick for Herbst (32), Stevenson for van der Merwe (79), Williams for Wilson (51).
Ref: Peter Fitzgibbon (Ireland)