Oxford City did Thame a favour by holding Ryman Division 1 promotion rivals Bedford to a goalless draw in front of their biggest crowd of the season at Court Place Farm on Saturday.
Mark Druce had an early effort beaten away by Heeps, and the Bedford keeper also kept his side on level terms at the break when he tipped over a chip from Matty Whitehead.
In the second half, City were unlucky when Dwaine Strong rounded Heeps only for Haley to hack the ball off the line onto a post before it rebounded into the Bedford keeper's hands.
With five minutes left, a looping header from Lawley struck City's bar.
Oxford City: Foster, Farley, Dark, Peirson, Smith, Whitehead, Lewis, Ferreira, Druce, Strong, Wimble. Subs not used: Hayward, Sweeney, Thorp.
Referee: E Mackrell. Attendance: 427.
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