Bristol's search for their first Aviva Premiership win of the season goes on but they came close to victory at Kingston Park.
Newcastle were 19-0 up at half-time with tries from Juan Pablo Socino, skipper Will Welch and Joel Hodgson and two Mike Delany conversions, but in the end they were clinging on for a 19-14 victory.
At the break it seemed only a matter of when the Falcons would clinch a four-try bonus win, but it was Bristol who dominated the second half, scoring two converted tries.
Mike Delany and Hodgson missed long-range kickable penalties which would have eased the Falcons' nerves as Bristol battered away in vain in search of that elusive win.
The Falcons started the match brightly and Mark Wilson, Ally Hogg and Scott Lawson all made strong drives before Socino struck in the fifth minute.
David Lemi lost the ball in the tackle and in the resulting scramble it was popped up and Socino stepped in to snap it up and score in the corner for a 5-0 lead.
Despite the loss of Alex Tait and Dominic Waldouck in the space of couple of minutes which meant a reshuffle of the Falcons back line, the hosts still looked very dangerous every time they moved the ball. Hodgson came on at fly-half, Delany moved to full-back and Belisario Agulla took up a role at centre.
Waldouck suffered a concussion collecting Jordan Williams' penalty rebound off the post and it was costly to both sides. Waldouck was helped off and Newcastle swept upfield to score their second try.
The home team hammered away with some excellent recycling and Sinoti Sinoti wriggled his way into the 22 where the Falcons pack took over. Calum Green and Lawson led the pick and drive up to the Bristol line where Falcons skipper Welch crossed and Delany converted for 12-0 in the 16th minute.
Another penalty miss, this time by Adrian Jarvis, was a disappointing return for a period of Bristol domination, and Hogg's swift breakaway up the blindside should have led to a try but Sinoti was stopped 10 metres out.
Nevertheless, Newcastle pounced from the scrum, with Delany the decoy at short side fly-half. Socino swept away on the open side and his pass to Hodgson took out three defenders for an exquisite try which had the crowd on their feet, and Delany converted for 19-0 in the 32nd minute.
Bristol dominated the opening 10 minutes of the second half using their forwards to pile drive through the middle and it paid off when Jon Fisher ended a 15-metre rolling maul with a try converted by Jarvis for 19-7 in the 51st minute.
It put Bristol right back in the game but Newcastle finally found something up front and won a scrum penalty, only for Delany to miss the kick at goal.
The pressure from Bristol returned and Mitch Eadie crashed through, with Newcastle constantly second best in contact, and a conversion from Jarvis made it 19-14 in the 62nd minute.
Hodgson missed a long-range kick at goal after Ben Glynn's late hit on Delany, and it was still anyone's game.
Bristol's Max Crumpton and Newcastle's Alex Rogers were sin-binned with five minutes left by referee Wayne Barnes for illegal scrummaging, and the remaining stages saw the Falcons win enough possession to close out the game.