The visitors took the lead on 14 minutes through Stina Blackstenius, but it did not take long for Leah Galton to level things up.
A packed-out crowd roared the Reds on throughout and, with time running out, substitute Melvine Malard took the lead for United her on WSL debut.
The striker's finish looked to be the winner, but a wonder strike from Cloe Lacasse saw the sides share the points.
FIRST HALF
It was a cagey affair in the opening minutes with both Arsenal and the Reds having early chances.