The first chance of the contest came after 18 minutes when Laura Bartup looked to trouble the goalkeeper from 25 yards out, but saw her effort flash wide of the goal. Bartup then came close twice in quick succession, firstly firing just wide before then striking over from 15 yards out.
But it was Chorley who took the lead four minutes before the break when Lisa Topping fired home from 25 yards out.
That goal was enough to send the Lancastrians in with the lead at the interval.
United kept pressing for an equaliser and Kathryn Smith was denied by a fine save from Bradley.
The Whites got the equaliser their play deserved when Smith fired home from inside the area. The attacking midfielder showed great skill to create space and the chance for herself before hitting the back of the net.
Leeds then had a great chance to take the lead when Rebekah Bass was felled in the area. The defender motored down the left-hand side and was brought down, earning her side a penalty.
But Danby saw her penalty saved by Bradley as the keeper continued to impress between the posts for Chorley. Both sides went for the a winning goal with Charlotte Evans coming close for visitors and Bartup shooting over for Leeds, but neither side could force a winner.
Leeds: Joyce, Bass, Hamill (C), Hannon, Robson, Danby, Hunt (Dobby 57), Smith, Williams (Underdown 73) Bartup, Soares-Williams. Subs not used: Wattam, Beacock.
Chorley: Bradley, Catlow, Searson, Holt, Topping, Walker, Stocks (Evans 71), Bell, Windle (A. Collins 77), Rotheram (Cleary 62), Hickson (L Collins 83). Sub not used: Matthews.