Joe O'Connor's goal with three minutes to go sparked a night of celebrations in the seaside resort by the club's 2,000- strong travelling army. Baldwin, who danced across the pitch at the final whistle, said: "I want Manchester United at Old Trafford.
"I have told my players it does not get any better than this. I had a game plan and it worked brilliantly. This is a dream come true for us. It is the greatest day in our history."
O'Connor tapped the ball in after Keith Russell and Richard Dandy had linked up to split the defence of a Blackpool side which had only been beaten once at home in the league all season. O'Connor said: "The goal has to be one of the best of my career, even though it was only two yards out."
Dandy, just 18, had been involved in two previous matches for the club last week, losing both of them - 6-1 and 10-3.
Hednesford will be joined in the draw by three other non-League sides: Woking, Stevenage and Enfield.
Rarely has victory by a non-League side over one of its supposed superiors been more expected than the 2-0 win of Conference side Woking at Cambridge United which saw the 39-year-old Clive Walker again on target plus one from Robbie Taylor. Walker had been the match winner at Millwall in the previous round.
Roy McFarland, the Cambridge manager, claimed it was a cross not a shot which put the skids under his team and said: "Clive Walker was aiming for somebody coming in at the far post, but the ball dropped over our keeper. That was what made the defeat all the worse, although we never really got going in the game."
Walker did not agree. "I noticed in the first half that the Cambridge keeper was off his line when a right-wing cross came over. So when I got the chance in the second half I decided to go for it and hit a deeper shot to the far post. Cambridge started hitting a lot of long balls and that suited us because we love to counter-attack."
The former England goalkeeper, Peter Shilton, suffered one of the lowest points of his career as Stevenage beat Leyton Orient 2-1 at Brisbane Road. Shilton was beaten by Corey Browne after only 54 seconds and, although Justin Channing equalised for Orient eight minutes later, Neil Catlin grabbed the winner for Stevenage.
Ten-man Enfield face a replay at Peterborough after forcing a 1-1 draw against Barry Fry's side. Enfield, managed by George Borg, an old friend of Fry's, went ahead through Shaun Marshall but Peterborough equalised through Ken Charlery after Al-James Hannigan had been sent off for the home side.
"We know we wasted the chance today but all credit to Enfield," Fry said. "They fought like tigers and deserve a second go just for that."