The Skrill Conference Premier team seized control from the first whistle and they almost opened the scoring in the 10th minute when Andy Todd saw an effort hit the bar after Tony Capaldi's free-kick.
Mark Wright fired one over after a strong run as Cheltenham struggled to make any impact against their non-league opponents.
Terry Gornell sent a shot over the bar from 25 yards in the 19th minute for Cheltenham, but Tamworth took a deserved lead seconds later.
Craig Braham-Barrett was dispossessed by Justin Richards and the ex-Cheltenham forward set up Chadwick, who applied a neat low finish past Scott Brown.
Cheltenham came to life in the final seconds of the first half, with Jamie Cureton missing a good chance after a clever exchange of passes with Byron Harrison.
Gornell then saw a header held by Cameron Belford at the near post, but Tamworth deserved their lead.
Belford turned substitute Steve Elliott's overhead kick over the bar in the 59th minute as Cheltenham showed a marked improvement, but they could find no way through and David Morgan, Richards and Andy Todd all went close to adding a second.