A foul on Thistle midfielder Anders Batarrita resulted in a penalty, but the Port Hill goalkeeper saved Ure’s spot-kick.
Up 1-0 at halftime, Thistle expected a wind-assisted onslaught from Port Hill but had the better of the first 15 minutes of the second spell.
Thistle man of the match Kane Stirton was brought down in the penalty area in the 60th minute and Batarrita scored from the spot to make it 2-0.
Port Hill came back into the contest with a goal in the 70th minute, the wind taking the ball beyond the Thistle defence.
Facing a revitalised team with a howling wind behind them, Thistle settled in for a rearguard action.
They looked to have done enough until those two last-gasp corners, the second of which resulted in a far-post header that beat the keeper.
When Port Hill scored midway through the first half of extra time, it looked all over as some of the Thistle players were out on their feet.
But two minutes into the second spell, Brad Reynolds met Ure’s in-swinging left-wing corner at the near post to score with a well-placed header.
Port Hill weren’t done, though. They scored with seven minutes of extra time to go, and — against the wind and on tiring legs — Thistle struggled to make an impact.
Late in the game, 21-year-old Mitchell Turner — who had a strong game on the right flank — had a chance but his shot hit the side-netting.
As well as midfielders Stirton and Turner, centrebacks Olly Tilley and Brad Reynolds shone in a Thistle performance that lacked nothing in spirit.
Gisborne United were among the teams who had a first-round bye.