The visitors scored four tries – through Matt Stevens, Martyn Wood, Zak Feaunati and James Scaysbrook – to kick-start their campaign a much needed boost.
Bath started the brighter of the two, with the hosts barely touching the ball before Olly Barkley opened the scoring with the simplest of penalties, after the Falcons came offside.
However, the England international missed his next three kicks – all from awkward angles on the left flank – to allow the hosts to establish an imposing lead.
Barkley, 23, came closest with his three first-half misses with an effort that struck the left upright and bounced to safety.
After his third miss Falcons officials asked for silence during kicks, to the ironic cheers from spectators.
Australia veteran Matt Burke, in contrast, coolly kicked the hosts level from a simple position after eight minutes when Bath were caught offside.
Burke, who also missed an ambitious penalty from the halfway line, relieved Dave Walder of kicking duties, but the England hopeful displayed the quality of his right boot by nudging the hosts ahead in the 20th minute.
The fly-half collected the ball 25 yards from the posts and executed his drop goal with perfect trajectory.
Burke extended the lead with another penalty, reward for his own surging run ending in an infringement. The opportunity had arisen after Matt Perry had dropped the ball in a position of promise, leading to a turnover.
The first try of the afternoon was also provided by Burke, the 32-year-old full-back, after he received a pass from Walder in the 38th minute.
There was still plenty of work for Burke, but he tricked two opponents with his dummy before scampering under the posts. He added to his points tally by converting.
Stevens, however, reduced arrears in first-half injury time when he stole in on the blindside, with Newcastle going into the break 16-8 ahead and with Harris in the sin-bin for an infringement in a maul.
With use of their extra player, Bath were back within a point of the hosts in the 48th minute through Wood after an advantage from referee Ashley Rowden.
Wood’s own pass hit Andy Perry in an offside position, and while players appealed the Bath player picked up the ball and scored. Barkley added the extras.
The visitors were ahead in the 58th minute when Feaunati was bundled over after a drive, with Barkley converting.
The bonus point was sealed for Bath when Scaysbrook crossed, although the 69th-minute try had much to do with the endeavour of France international David Bory, who made vital ground in opposition territory.