Press had a quiet afternoon, often finding herself isolated up front, and was substituted after 55 minutes while Heath registered an assist for Turner's goal.
It was a frantic first half with neither side finding the back of the net.
The best chance of the half came within the first six minutes with Tottenham's Rosella Ayane just failing to convert. United tried to play the ball out from the back but instead it found its way straight to the striker's feet who pulled her shot wide.
The deadlock was finally broken on 66 minutes by Turner who rose above the Spurs defence to get her head to a perfectly placed Heath corner which she looped over Spencer.
It was an important win for Casey Stoney's side who crashed out of the Continental Cup on Wednesday when they lost 3-1 to Championship side Liverpool.
"[I am] relatively pleased. I am always a realistic. I don't think we started well enough," Stoney said after the game.
"I am pleased for Millie as she had to turn it around as she knows she had to put in a performance after Wednesday.
"What we needed to today was a win and we did it."
Saturday's result means United have secured three wins and a draw in their opening four games, enough to put them top of the WSL until at least Sunday when Arsenal and Everton have the chance to overtake them.