Kiwi fly-half Jimmy Gopperth tormented the home defence with his kicking out of hand, setting up the clinching try for full-back Charlie Amesbury to score his second of the game with a clever dink off the outside of his foot.
Half-back partner Micky Young was just as much of a handful, scoring the try of the game on the hour with an 80-metre intercept and James Hudson also crossed for the visitors.
Nicky Little kicked Bath ahead before, almost straight from the restart, Bath added to their lead with a try by Matt Carraro. The ball was turned over at a ruck on the visitors' 22, and Pieter Dixon fired out a long pass to the Australian centre - who squeezed in at the corner for an unconverted try.
Gopperth was wide with another penalty on 34 minutes. But just before the break, Little failed to hold Amesbury in a tackle - and the full-back wriggled away to score in the corner under a pile of bodies.
The game suddenly came to life after the restart, Abendanon's stunning counter-attack creating a try for captain Michael Claassens after good support by Julian Salvi and Carraro.
But Little was again off target - and within three minutes Newcastle had cancelled it out with an unconverted try of their own, scored by Hudson.
The home team then imploded, conceding 15 points in 10 minutes.
First, Young intercepted a pass from Dixon in his own 22 to run in for a try converted by Gopperth - and although an increasingly desperate Bath responded with a drop goal by Ryan Davis, Gopperth kicked a penalty and then slid a delightful grubber kick through for Amesbury to score.