Ward-Smith was one of the game's two try scorers
Bristol: (8) 11
Pens: Strange (2)
Dragons: (0) 7
Try: A Brew
Ex-Newport outside-half Jason Strange booted the Dragons to Challenge Cup defeat at the Memorial Stadium.
The Welshman kicked a penalty in each half as Guinness Premiership leaders Bristol edged a scrappy, rugged match.
Bristol back-rower Dan Ward-Smith bagged the only try of the first half, but in-form Dragons winger Aled Brew cancelled it out after the break.
Ceri Sweeney's conversion cut the gap to one and - despite Strange's second kick - the Dragons got a bonus point.
Bristol head coach Richard Hill:
"We were a bit sloppy last weekend, but we still won and it gave me a chance to read the riot act to them.
"Playing the Dragons was probably our hardest physical challenge this season. So I was pretty pleased.
"There was a danger a bit of complacency was creeping in but we were much better."
Bristol: C Morgan; Stanojevic, Higgitt, Cox, Lemi; Strange, Perry; Clarke, Regan, Hobson; Winters, Llewellyn; Budgett, To'oala, Ward-Smith.
Replacements: O'Riordan for Perry (40), Hilton for Clarke (79), Thompson for Hobson (51), Short for To'oala (44). Not Used: Blaney, Gray, Arscott.
Dragons: Morgan; N Brew, Emerick, Smith, A Brew; Sweeney, Cooper; Black, Daly, Harrison; Gough, Charteris; Owen, Charvis, Fitisemanu.
Replacements: Fussell for A Brew (73), Crawford for Daly (49), Corsi for Gough (53), Sidoli for Charteris (50), Hall for Fitisemanu (35).
Not Used: Walker, Hercus.
Sin Bin: A. Brew (36).
Referee: Eric Darriere (France).