A strong start for the visitors was rewarded when Dora Zeller struck in the 56th minute, but Maxwell's powerful strike levelled the score in the 79th minute and they pushed for a winner with Simone Magill going close twice in the late stages.
A quiet start soon livened up when Sarah McFadden's shot from a corner was saved, but the rebound allowed Emoke Papai to burst through the home defence and Shannon Turner made a solid save.
Zeller proved to be a threat for Hungary, coming close twice.
The visitors threatened again as Henrietta Csiszar crossed the ball to Evelin Fenyvesi, but Turner was equal to it and Northern Ireland quickly had a good effort at the other end when McFadden sliced the ball narrowly wide.
However, Hungary took the lead in the 55th minute with a stunning corner from Zeller, as her set-piece beat everyone in the six-yard area to curl into the net.
The hosts then found the equaliser in the 79th minute with an excellent strike from Maxwell, who cut just inside the box and fired past Reka Szocs.
The hosts pushed for a winner after a brilliant pass from McFadden found Magill down the centre, but the forward blasted the ball wide and had another great chance smothered by Szocs just moments later.