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Bristol's Cook Islands international Tommy Hayes opened the scoring with an early penalty goal and the conversion of Ward-Smith's try, with home fullback Jaques Schutte kicking two penalty goals to keep his side in touch.
Bristol centre Rob Higgitt scored to give his side a 17-6 lead at the break, but they lost Argentinian second-rower Mariano Sambucetti to the sin-bin for a cynical foul.
Ward-Smith and winger Lee Robinson bagged second-half tries to give their side a match-winning lead which was dented by late five-pointers from front-rowers Matias Aguero and Luigi Ferraro.
The victory takes Bristol above Narbonne and Northampton Saints in the table, and home wins over Narbonne and Viadana in the closing rounds should be enough for a spot in the quarter-finals.
In Friday's other Challenge Cup match, Montpellier thumped Amatori Catania 74-12 at Stade Sabathe, with fullback Frederic Benazech bagging four of the French side's 12 tries as they moved off the bottom of Pool 5.
