Buxton stay third in the NPL Premier after beating the second-placed outfit on Saturday, despite losing key players Diego De Girolamo and James Hurst through injury in the first half.
The winner came from substitute Ash Chambers after 78 minutes, in front of over 2,000 spectators, and Hayward praised his team for their strength of character and hard work against the full-time professionals of South Shields.
He said: "It was a really good team display. We set up really well and executed the game plan perfectly and caught them on the counter. This is a side that doesn’t know when it’s beaten and it’s a great result for all at the club.
"It was a really enjoyable weekend overall as we’d travelled up on Friday and stayed over, then had music pumping on the bus all the way home.
"Losing Diego and James early on was obviously a blow but to get the result despite that shows the quality we have. We thought we’d be in more trouble but did well.”
Hayward paid tribute to match-winner Chambers, who again came off the bench to score.
He said: “The whole team have been fantastic but Ash has been breathing down Diego’s neck all season and wants to play. It shows what quality we have in attacking areas and we may still look at further additions, although centre-half is also a priority with Ben Turner out.”
Saturday's goal scorer Chambers took his goal calmly, latching on to a loose ball and lifting the ball into the roof of the net. He said: "I had a couple of chances earlier, so I was relieved to make sure with this one. It was a great result for all the lads against a really tough team."
Following the FA Trophy game with Colne this weekend, the visit to York City on November 6 will mark the Bucks' first match in the FA Cup first round proper since season 1962/63.
Tickets are on sale at www.yorkcityfootballclub.co.uk, £12 for adults, £6 concessions and under-5s free. There are plans for a family coach and supporters' coach from Buxton, bookable via www.buxtonfc.co.uk.