For the third time in four games City ended the match with 10 players and for the second time in a few days played the entire second half with 10 men as Clayton McDonald was sent off for a second yellow card.
Moors dominated the game forcing Nathan Vaughan into several excellent saves
Darryl Knights hit the post from 20 yards before Vaughan pulled off an outstanding one-handed save to keep out a point-blank Knights' shot.
Ryan Beswick's long range effort was parried away by Vaughan before the Moors finally found the net on 33 minutes only for goalscorer Asante, who was stood on the line to be flagged offside.
On their next attack Moors broke the deadlock. Clayton McDonald, who had earlier been cautioned was given a second yellow for handball and sent off. Stefan Moore, scoring from the spot.
Three minutes later is was 2-0 from Knights far post shot after City failed to clear from a throw-in.
On the stroke of half time it looked certain to be 3-0, but Vaughan again denied Knights with another acrobatic save.
The second half was much the same with Vaughan making several saves, one from Ryan Beswick, tipping over the bar.
On 63 minutes Jack Byrne should have added another but his shot went wide of the post.
City had a late shout for a penalty when substitute Jamie Spencer appeared to be brought down in the box.
Vaughan made two further saves from Darryl Knights before in the 82nd minute could do nothing as Moors added a third from close range.
Att: 525