It looked as if Bath were going to suffer yet another home defeat when Anthony Elliot touched down in the corner from a chip ahead to give his side a 13-8 lead.
But Best squeezed into the corner with minutes remaining to level the scores and Chris Malone landed the conversion from a difficult angle.
Overnight results had left Bath back in 11th place but this result allows them to leapfrog Saracens and Bristol.
Bath took the lead inside five minutes with a well-worked move which saw winger Salesi Finau barrelling through for a try under the posts. Malone kicked the conversion and added a penalty from short range on 18 minutes but it was not all plain sailing.
Newcastle had their moments and Alex Crockett had to be alert to haul down flanker Ben Woods after he broke through in midfield.
On at least two other occasions the Falcons lost promising opportunities with poorly timed passes. Their best opportunity came when wing Ollie Phillips cut through only for Woods to spill an admittedly difficult overhead pass in the shadow of the posts.
However they continued to raise the pace and were rewarded with Matt Burke’s penalty from close range on 23 minutes.
Burke had a more difficult chance just before the break but was only just wide from 38 metres.
However, when he was penalised for not releasing in the tackle soon after half-time Malone kicked his second penalty to extend Bath’s lead to 13-3.
Home scrum-half Nick Walshe was beginning to find some holes but then blotted his copybook by getting sin-binned for slapping the ball out of James Grindal’s grasp. Burke kicked the penalty on 51 minutes.
Bath hung on with 14 men for another nine minutes before Mark Mayerhofler created space for Phillips finally to score in the left corner, Burke adding the conversion to bring Newcastle level at 13-13 on the hour.
Newcastle suddenly looked full of running and both Phillips and Mayerhofler were held up short of the line. Bath were looking tired and short of ideas and it was no real surprise when Elliot scored his try on 71 minutes. Bath gathered up a final burst of energy though to secure the points.