Aiden Savory put the Brewers ahead after just four minutes before the dismissal of Milne was quickly followed by Paddy Miller equalising.
The constant pressure from the Tigers eventually led to them taking control.
Karl Jones nudged then ahead on the hour mark before goals from Matt Beadle and James Burke added gloss on the victory.
Manager Michael Morton made a single change to the draw over Trafford as Aiden Savory passed a fitness test and was back in the starting line-up.
The former Farsley man had gone a couple of months without a goal after a potent start to the campaign but he was back on target again inside five minutes.
Seconds after Andy Milne recovered to deal with a dangerous back pass from Chris Howarth, the ball was in Hyde’s net.
Corey Roper’s sweeping ball in behind led to keeper Pete Crook rushing off his line but this only caused confusion in the back line which allowed Aiden Savory to nip in and slide into an empty net.
It was a frantic start to the game as ten minutes hadn’t even passed when Matthew Beadle rounded Michael Ingham and went to ground wanting a penalty, only for the referee to award a goal kick.
The home side were hungry to respond immediately as Harry Coates marauded forward and tested Ingham’s handling.
Hyde continued to put pressure on Taddy and cane close on fifteen minutes when Porritt fizzed into Beadle who cryuff flicked first time to Tom Pratt but he bent wide.
Having said that, the Brewers rode this mini wave from the Tigers and order was relatively restored for a period.
Morton’s side threatened to extend their lead shortly after the half hour mark as Aiden Savory’s strike from outside of the box was tipped wide by Crook before Greening whipped on target.
Nevertheless the turning point in the contest came two minutes later as skipper Andy Milne was shown a straight red card.
The defender was adamant that Tom Pratt was in an offside position as the ball was played through but the flag didn’t come and Milne brought down the attacker.
This saw versatile man Aiden Savory move into the back four to partner Josh Barrett.
Janni Lipka sent a fierce strike into the side-netting as Tad looked to have made it to the break with their lead still intact.
However, former Brewer Paddy Miller levelled right on the stroke of half-time as he flicked onto his weaker right-foot before his drive deflected in off the underside of the crossbar.
Half-Time: Hyde 1-1 Taddy
Taddy came out in the second forty five and looked to frustrate the Tigers by sitting in, getting bodies behind the ball and making it difficult for them to find space.
And this was working for the first fifteen minutes of the half as Harrison, Burke and Jones were reduced to long-range strikes.
However as the clock flicked over the hour mark, the home side nudged in front and effectively killed off the contest.
First-half substitute Karl Jones somehow got beyond Hunter and Murray before sneaking a show into the bottom corner.
When Albion did regain possession, they had very few options going forward with Tom Corner isolated up top.
Josh Greening cleverly tried to catch Crook off guard by going for a spectacular flick up and volley from all of 35 yards.
It was well executed by the exciting attacker but not enough loop on the effort to trouble the former Harrogate Town stopper.
There’s no doubting that Hyde are a very strong side in all areas of the pitch, but Janni Lipka was impressing the most in central midfield.
If the result was still somewhat in doubt, it was rubber stamped with fourteen minutes remaining.
Hyde created an overload down Albion’s right-hand side and the low cross was turned in at the near post by Matt Beadle.
This goal definitely killed off the contest and Darren Kelly’s side grabbed a fourth in the final minute.
Dominic Marie’s miscued shot dropped kindly for James Burke to tap in from just a couple of yards.
Full-Time: Hyde United 4-1 Taddy
Hyde: Crook, Harrison, Burke, Lipka, Coates, Miller©️, Sutherland (Jones 31), Churchman, Beadle, Pratt (Boyle 82), Porritt (Marie 54)
Unused: Brown, Gay
Tadcaster: Ingham, Howarth, Hunter, Roper, Milne©️(SENT OFF 35), Barrett, Murray (Davidson 81), Sellars, Corner, Savory (Mycoe 55), Greening
Unused: Hardy, Akeroyd, Stewart
Att: 383