Oxley made two fine stops to deny Mo Eisa inside the first 10 minutes before later keeping out Daniel Harvie’s strike from range as the hosts applied early pressure.
Having ridden the storm, Harrogate went ahead in fortuitous circumstances when Dons goalkeeper Craig MacGillivray’s clearance struck the unfortunate Warren O’Hora and flew into the back of the net four minutes before the break.
The visitors dug deep in the second half to secure all three points and leave their manager beaming.
“We are buzzing and delighted with the victory,” said Weaver.
“We initially lacked a little bit of the intensity which we really desire and demand. It probably took a fine save from Mark Oxley to be the catalyst to have a better performance.
“After the first 15 minutes we were terrific and really hard to beat, which we have been for the last few weeks.
“The confidence has grown with our efforts on the ball in order to counter-attack and pass our way up the pitch, and we showed great courage in that respect at a difficult place to come.
“There were signs on the counter-attack, where we caused a bit of mayhem, sucked them into the middle third, and when we broke it up, we had threats.
“Second half was all about grit and seeing it through.”
Milton Keynes Dons: MacGillivray, O'Hora (Grant 68), Stewart (Tucker 46), Harvie, Tomlinson, Payne, Williams (Leko 69), Gilbey, Ilunga, Harrison (Dean 81), Eisa. Unused substitutes: Robson, Dennis, Kelly.
Harrogate Town: Oxley, Ramsay, O'Connor, McDonald, Gibson, Falkingham (Burrell 82), Sutton (Cornelius 75), Thomson (James Daly 81), Daly, Odoh, Folarin (March 65). Unused substitutes: Thomas, Muldoon, Foulds.
Referee: M Coy (County Durham).