Josh Greening got the Brewers off to a good start to open the scoring before Conor Sellars came off the bench to immediately cancel out Tommy Wood’s equaliser with a screamer.
Michael Morton made three changes to the loss over Skelmersdale with recalls for Michael Ingham, Jason Mycoe and Corey Roper with Adam Baker on the substitutes bench.
On a windy afternoon in Yorkshire, Taddy almost got off to a perfect start inside the very first minute as Tom Corner saw his header flick off the post.
It was a fairly even opening exchanges as Michael Ingham, who was making his first start in six weeks, was first called into action after ten minutes making a good save to deny Sanchez Payne.
But it was Taddy who struck first just short of the quarter of an hour mark.
High pressure from a combination of Josh Greening and Tom Corner saw the latter play in the former and Greening carefully volleyed past Souter and into the back of the net.
Former Scarborough forward Leon Osborne thought he had a clear shot at Ingham’s goal ten minutes later but Josh Barrett swept across to make a superb interception.
Tom Corner had two chances come and go in quick succession on the half hour mark, one considerably better than the other.
His first came from an Ingham goal kick with Murray nodding the towering forward in behind but he dragged wide of the mark.
With chances somewhat drying up with the half reaching it’s conclusion, the Brewers were forced into making a change with Pete Davidson pulling up with what looked like a reoccurrence of his earlier hamstring problems.
Conor Sellars quickly got stripped and entered the field with a minute to go.
There was one final chance in the first half for Morton’s side as Tom Corner volleyed inches wide after good play from Corey Roper and Chris Howarth.
Half-Time: Taddy 1-0 Ossett Albion
Into the second half and Tadcaster really should have been out of sight after squandering two clear cut opportunities inside the first five minutes.
Sellars threaded a precise through ball to an unmarked Josh Greening but opted to try pick out Corner instead of going alone but Ossett read it.
Corey Roper, who was having a terrific game in the centre of midfield, saw his low strike from distance spilled by Souter, Greening followed up but the keeper made amends and was keeping his side in it.
The Unicorns threatened for the first time in the second half ten minutes in and were literally cats hair away from equalising.
Aidan Chippendale cut inside and curled a beauty towards the far corner only for Ingham to get the faintest of touches on the effort which saw it incredibly strike both posts.
Next it was Aiden Savory’s turn to have a go and he came increasingly close to doubling Tad’s lead when his diverted effort was somehow kept out by Souter.
Nevertheless the Unicorns stayed within touching distance and levelled with twenty minutes to go.
Matt Dempsey’s free-kick was nodded into the back of the net by number nine Tommy Wood.
But the two sides were level for less than a minute as substitute Conor Sellars sent a rocket past Souter from 22 yards to win it for The Brewers.
Taddy should have made the final minutes more comfortable but Souter again was the thorn in the home side by keeping out Corner from close range.
The final moment of real controversy came five minutes from time when the former Swansea City striker Jason Price controlled and sent a brilliant half volley into the net only to be adjudged to have handled the ball.
As Ossett pushed everyone up, Aiden Savory was played clean through to seal it but yes you guessed it, Souter saved.
However, it didn’t matter as that was that.
Full-Time: Tadcaster Albion 2-1 Ossett Albion
The Brewers now travel to Cleethorpes Town in the Second Round of the Integro League Cup.
Tadcaster: Ingham, Howarth, Hunter, Davidson (Sellars 44), Mycoe, Barrett, Murray (Baker 75), Roper, Corner, Savory, Greening (Milne 89)
Unused: Hardy
Ossett: Souter, Dawson, Hardaker, Wells, Wilkinson ©, Dempsey, Chippendale, Mbeka, Wood, Payne (Price 69), Osborne (Bates 81)
Unused: Hurtley, Kabeya, Wright
Att: 175
Ref: Luke Watson