Having trailed 2-0 with 17 minutes remaining, Ross Hannah pulled one back before substitute Matty Waters levelled in injury time with a superb strike from the edge of the box.
But moments after restoring parity Chester hearts were broken when a corner wasn't dealt with and eventually crept over the line via a deflection from goalkeeper Conor Mitchell.
It was rough on the Blues who thought they had done enough to earn a share of the spoils on a bleak and wet night in the North East.
"The boys are absolutely spent, absolutely on the floor, they've given it everything - beyond everything," said Shaw.
"I'm on the sideline thinking that the fans back home will be thinking that we've not come and done it at Gateshead and we're 2-0 down but I'm ready to come and talk to you and hope the fans who have come - and it's amazing that they have come up here - tell their mates how well we played.
"They fought back, they put their bodies on the line. They have given it everything for the cause and everything for me. I'm just a 30-year-old who has never done it before, on a Tuesday night at the bleakest ground possible and they could easily have not been at 100% but they have come and given it everything. It makes me proud."
Shaw takes the reins once again this weekend for the trip to Dover Athletic on Saturday (3pm) while Chester's search for their next manager hots up, with the application deadline closing today and the new man set to be in place for the visit of Maidenhead United on Saturday, September 23.