Boro made two changes for the cup tie, with Jackson Jowett coming in for his debut and Will Annan moving out wide on the other flank.
The first half was a drab affair, with the positive football played being led by the hosts, who were looking for back-to-back victories.
Things came alive the longer the period went on, with Boro creeping closer and closer to Michael Ingram's goal.
Their persistence finally paid off with eight minutes of the half remaining, when good work in approach by Michael Coulson and Walshaw ease in Matty Dixon to drill into the top corner.
Tadcaster almost responded immediately, but Coulson did his defensive duties when clearing a Harry Coates header off the line.
Right on the interval Ingram produced an immaculate save to push away Jowett's drive, then Coulson saw a deflected effort roll just past the Tadcaster post.
The second half was more like it, with Jowett trying to kick things off with a bang, but his volley was blocked on the line.
Things took a turn for the sour five minutes in, when Jack Walters went straight through Luke Dean in horrible fashion. The Tadcaster man was given a straight red and an injured Dean was replaced by Cadman.
The sub's first act was to play a delightful one-two with James Walshaw and then rifle Boro ahead with a sweet finish.
Tadcaster were not done though and just after the hour a tangle between Nathan Valentine and Lamin Colley resulted in the referee pointing at the spot. Aaron Hardy stepped up and beat Tommy Taylor with a neat finish.
Boro licked their wounds for a time and then pounced to reclaim their two-goal cushion.
Once again it was Cadman, who collected after a spell of pressure and gave Ingham no chance with a sweet drive.
Boro had the opportunities to kill things off, with Will Annan finding the side-netting and Walshaw seeing his effort pushed over by Ingram.
Tadcaster continued to show their spirit though in a game that many would have seen as already dead.
And to set up a grand-stand finish, Billy Whitehouse crept through a static Boro defence and rolled home to make it 3-2.
Rather than a thrilling climax though, the game tailed off and it was Boro who made their way through to the next round.