The Falcons led 14-8 at half-time before Bath pulled away with tries by Olly Woodburn - his second of the game - a penalty try and another from Samoan prop Anthony Perenise.
Young fly-half Tom Heathcote celebrated his 20th birthday with two conversions and a penalty and had a hand in three of the tries, while Olly Barkley scored the rest of Bath's points.
A late Will Chudley try, converted by Jimmy Gopperth, ensured Newcastle also took a bonus point.
Tenth-placed Bath entered the game in high spirits after three wins on the bounce, but New Zealand fly-half Stephen Donald, hobbling around in a knee brace with strained ligaments, was a notable absentee.
It was less than two minutes before Barkley struck a 40-metre penalty to open the scoring.
Newcastle saw very little of the ball for the next 10 minutes as Bath strung the phases together in the winter sunshine.
When the Falcons did build some pressure, the ball was turned over and Ryan Shortland conceded a penalty with a late hit on Tom Biggs.
Bath returned to the attack and Heathcote worked a neat loop with Barkley to send Woodburn in at the corner for a 15th-minute try. Barkley was wide with the conversion.
The riposte was immediate as Michael Claassens fired a pass above Barkley's head in his own 22 and the Falcons swooped on to the loose ball.
Referee Dean Richards was still signalling penalty advantage as number eight Mark Wilson scored on the overlap wide on the left. Gopperth converted to leave his side just a point adrift at the end of the first quarter.
Ten minutes later they found themselves in front after Woodburn struggled to deal with a teasing kick by Gopperth and the Falcons pack set up camp on the Bath line. Spurning a penalty, they opted for another scrum and hooker Rob Vickers forced his way over, leaving Gopperth to put the Falcons 14-8 ahead.
The visitors suddenly oozed confidence while Bath lacked composure and began to make unforced errors. Heathcote had a chance to pull back three points on the stroke of half-time but his 48-metre penalty grazed the underside of the bar.
More sloppy play by Bath on the restart could have cost another try as Gopperth prodded the ball into the corner but Sam Vesty was there first and shepherded the ball into touch.
It was 10 minutes before Bath regained control but they then scored two tries in five minutes. The first was created by a marvellous piece of skill by Perenise, breaking clear and throwing a dummy before slipping a perfectly weighted pass into the Falcons' in-goal area where Woodburn won the touchdown.
Heathcote did not convert but could not miss the next one on 55 minutes when referee Richards awarded a penalty try. Taking a leaf out of Perenise's book of tricks, Heathcote had nudged the ball to the try-line for Biggs only for the winger to be tackled early by Gopperth.
Bath stepped up the pace in search of the bonus point, which finally came in the 65th minute when Perenise was bundled over in the corner. Heathcote converted from the touchline and then swapped penalties with Gopperth to maintain his side's 13-point advantage.
But Newcastle launched a late rally to take a bonus point of their own as with the last move of the game, replacement scrum-half Chudley scrambled over in the corner, leaving Gopperth to kick the conversion.
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