Town take a point off leaders
They might have had all three points when Simeon Howell headed against the bar after Highmoor had Justin Forde sent off after collecting his second yellow card.
The leaders also had a late chance when Chris Rackley kept out Sanchez Commack's point-blank header, while the best chance of the first period fell to Highmoor's Danny Lingwood, but the defender shot over from close range.
Thatcham are back in league action on Wednesday night when they go to Wootton Bassett Town.
made cup progress with a 2-1 success at Easington Sports in the Bluefin Sports Challenge Cup.
They trailed to Andy Stidder's 25th minute penalty before Matt Pedder levelled the scores and Callum Willmoth netted the second half winner.
Newbury have more cup action on Tuesday night when Oxford City Nomads are the visitors in the Flying Press Floodlit Cup.