The visitors made a fast start as a second minute corner was cleared to Henry Sady who fired home from 20 yards.
Sady advanced on goal but was denied a second as Thatcham keeper Matt Pullinger dived at his feet, and Sady had a goal disallowed towards the end of the first half, as did Christian Johnson for the Kingfishers.
In the second half Mo Ceesay fired just off-target for the Kingfishers from the centre of goal, then Callum Willmoth tested Jude Franks from distance. For Melksham substitute Elliot Haffenden curled one narrowly wide, and then found himself in acres of space on the left following Dawid Regula’s ball but fired over.
When Johnson flicked one into the side-netting it seemed that time had ran out on Thatcham. But in injury time Willmoth’s free-kick was knocked back into the danger area by Dan Smith and Oli Bourne bundled home the leveller from close range.
Next up for Yash Romeo's side is a trip to Bideford on Saturday.
Team : Matt Pullinger, Ben Portbury, Kyle Daniel-Spray, Callum Willmoth, Jacob Wilson, Mo Ceesay, Christian Johnson, Jadyn Dundas, Harvey Flippance, Sam Tanner, Jamie Davies. Subs on : Oli Bourne, Dan Smith, Lewis Slade, Jenson Mansel.