Skinner's side took the lead just before half time, courtesy of Ella Toone. United's no.7 was at the back post to meet Hayley Ladd's looping cross, slotting into an empty net after Manuela Zinsberger missed her attempt at intercepting it.
Frida Maanum levelled for the hosts immediately after the break, as her effort took a significant deflection off Maya Le Tissier to beat Mary Earps. The Gunners took the lead with 20 minutes to go after Laura Wienroither finished off Katie McCabe's sublime cross.
Just when it looked like Arsenal may collect all three points, Millie Turner's thumping header from Katie Zelem's free-kick pulled the Reds level.