The win keeps Bath in the race for Heineken Cup qualification, while the losing bonus point may well be enough to stave off relegation for the Falcons.
Newcastle were lucky to survive two breaks by Matt Carraro in the opening five minutes, as he surged through the middle. Only lack of support and some decent Newcastle defence prevented a try on both occasions.
It still left Bath in command at scrum 10 metres out and with a penalty when the Falcons dropped the scrum - but Butch James missed what was a straightforward chance, his kick drifting just wide on the stiff breeze which favoured the West Country side.
James Hudson's lineout take led to a an 11th-minute penalty when Bath dragged down the maul but Jimmy Gopperth had no more success than James with a scruffy strike which was nowhere near.
Newcastle wasted a couple of real chances with Alex Tait cutting the Bath midfield defence to ribbons but all they had to show for it was a 23rd-minute penalty from Gopperth.
The lead did not last long, with some excellent Bath ball retention and a typical Michael Claassens (pictured) try from close range making it 5-3 to the visitors in the 26th minute, with James missing another simple kick.
A silly obstruction on Micky Young led to Gopperth's second successful penalty in the 33rd minute and Newcastle led 6-5 in the 33rd minute.
However, the Falcons had no answer to the power of Bath's forwards two minutes before half-time other than to concede a penalty which James kicked for an 8-6 Bath lead at half-time.
Bath made a good start to the second half with Claassens prompting and probing and almost wriggling through for a second try.
Newcastle were very much on the back foot and were caught offside for Sam Vesty to edge Bath further ahead in the 51st minute, as England prop Davey Wilson was helped off after lengthy treatment and replaced by Duncan Bell.
The Falcons simply could not get out of their own half despite the wind advantage until Manning rattled through down the right - but Bath's solid defence made sure he made it no further than the 10 metre line.
The closing quarter was all Bath and Newcastle simply could not get their hands on the ball, with only a spill by Ignacio Fernandez Lobbe ending a series of effective drives.
However, a score was inevitable and it came from the boot of James in the 72nd minute - but only after he had missed yet another simple kick.
At 14-6, Newcastle were without that bonus point but then came the pivotal moment of the match when Tane Tu'ipulotu broke away and kicked downfield for Luke Fielden to win a frantic foot race and score the try which could keep the Falcons up.
Gopperth's conversion agonisingly hit the post with four minutes left on the clock but at 11-14, the Falcons had that crucial losing bonus point in their grasp and they nervously played out the closing seconds.