Arundell scored his four tries within 61 minutes of action at the Recreation Ground to send the champions just four points behind league leaders Northampton.
Bath played a quarter of the game down a man after hooker Tom Dunn was given a 20-minute red card for a head-on-head collision with opposite number George McGuigan.
Newcastle, owned by Red Bull, suffered a 15th defeat of the season out of 16 games and remain bottom of the table.
"We've qualified for the semi-final with two rounds to play, so that's really pleasing," Bath head coach Johann van Graan told the BBC.
"I thought we managed the red card very well and we scored some fantastic tries through the 80 minutes."
Bath led 31-0 at the break, after 23-year-old Arundell scored twice within the opening quarter.
Arundell doubled his tally for the game and took his count for the season for club and country to 22 tries by the 72nd minute to secure victory.
Elsewhere, Saracens moved three points off the play-off places with a 30-7 win over Gloucester and Harlequins came from behind to beat Exeter 41-24.
Despite defeat for the Chiefs, who led 24-14 at the break, they remain in the final play-off berth with Sarries breathing down their necks in fifth place.