It was all about the three points, and an early goal by Matt Wright and a second half penalty from Ed Palmer were enough to give City a fourth straight win and a ninth in their last 11 Evo-Stik Southern premier division outings.
They remain fourth, but are now level on points with the two teams above them – Weymouth and Corby Town – ahead of this Saturday's visit of struggling Burnham.
There were also two penalties, with City keeper Charlie Burns saving Ashan Holgate's spot kick at the death to ensure a clean sheet.
This City side are always up for a scrap and they never allowed themselves to be bullied by the physical Wiltshire visitors.
And the home side could hardly have got off to a better start.
But it was one of the rare moments of quality play in the opening 45 minutes as City were unable to build on the goal, though Hayles almost made it 2-0 when his shot from the right of the area was deflected just wide after he rounded Kent Kauppinen in the visitors' goal.
City defended manfully, but needed to improve their attacking game, and the introduction of Vassell and Sullivan for Hayles and Afful early in the second period certainly helped.
Wright was needlessly brought down from behind as the ball looked to be heading out for a corner and Palmer put away the spot kick for his first City goal, although Kauppinen almost kept it out as he dived to his right.
There was even time for Burns to save a Holgate penalty at the death after Arran Pugh had been penalised for handball.
CITY: Burns; Farkins, Palmer, Pugh; White, Krac, Ash (capt), Afful (Sullivan 52), Greenslade; Hayles (Vassell 52), Wright (Duff 69). Subs (not used): Green, Moore.
CHIPPENHAM TOWN: Kauppinen; Smith, Osman, Coupe (capt), Tindle, Preece, Ferguson (Phillips 71), Roberts, Holgate, Ballinger (Guthrie 90), Guthrie, Subs (not used): Francis, Pook, Harvey.
Red card: Tindle (two yellows).
Follow us: @westbriton on Twitter | westbriton on Facebook