Bristol winger Brendon Daniel weighed in with a hat-trick of tries, while Newcastle were forced to rely on the boot of their winger Liam Botham, who was successful with four penalty goals.
Botham struck two first-half penalties, but two tries from Daniel, together with one penalty and one conversion from Argentina Felipe Contepomi, saw the visitors hold a 15-6 lead at half-time.
Botham kicked two more penalties in the second stanza, to see the scoreline at 15-12 in Bristol's favour, before the 25-year-old Daniel completed his hat-trick in the final quarter to stretch his team's lead to 20-12.
Bristol have now scored 16 points and 16 tries in their last four matches, while Newcastle find themselves firmly entrenched at the bottom of the Premiership table.
For Newcastle Falcons:Pens: Botham 4
For Bristol Shoguns:Tries: Daniel 3Con: ContepomiPens: Contepomi