Bristol virtually secured Guinness Premiership status for next season as they moved to eighth place in the table by completing a double over the Falcons.
The home side looked to have made the ideal start after just two minutes from a turnover ball on their own 10-metre line.
Scrum-half Shaun Perry fed lock Gareth Llewellyn who put inside centre Sam Cox away on a break.
Cox passed out to right wing Lee Robinson who chipped over the head of an advancing opponent and collected the awkward bounce before powering over the line.
But international referee Chris White ruled that Robinson had been held up over the line by left wing Joe Shaw.
From the ruck after the five-metre scrum, Bristol were penalised and the Falcons cleared their line.
The first points came after 10 minutes when fly-half Jason Strange gave a long pass out to left wing David Lemi on the Falcons 22 and the Samoan international evaded a tackle to race down the touchline and score in the corner.
Strange added the extras from wide on the left and, seven minutes later, nailed a 30-metre penalty from in front of the posts after the Falcons had killed the ball at a ruck.
Bristol maintained their pressure until Falcons full-back Matthew Burke broke through to score from close range.
Fly-half Toby Flood added the conversion but Bristol hit back immediately. Strange, with the wind at his back, landed his second penalty – again from 30 metres – to give the home side a 13-7 interval lead.
The Falcons, perhaps looking for some added inspiration, changed their shirts at the interval in the hope of avoiding a second defeat to Bristol this season.
Despite some early second-half pressure from the visitors, Bristol struck first when Strange kicked his third penalty after the Falcons failed to release the ball at the tackle.
With the hour mark approaching, Strange gave a long pass to outside centre Rob Higgitt who flipped the ball to full-back Vaughan Going. Despite the ball appearing to go forward, the referee allowed play to continue.
Going fed Lemi, who had received a late call-up after Brian Lima withdrew, and he squeezed down the touchline to score in the corner – with Strange again on target with his conversion.
With six minutes to be added on, Bristol went in search of more tries, and a bonus point, but the Falcons defence showed late resistance.