The hosts trailed 24-17 with 13 minutes remaining at Kingston Park, but mounted a stirring fightback to claim their maiden win this term and move five points clear of bottom club London Welsh.
Scores from Thomas Waldrom, Ben White and Henry Slade had put Exeter in the ascendancy at half-time, although three penalties from Juan Pablo Socino and a try from Sinoti Sinoti kept the home side in touch.
Gareth Steenson kicked three goals to extend the Chiefs' advantage, but two tries in three minutes from Tom Catterick and Noah Cato saw Newcastle prevail.
Welsh are now the only side without a win, after their 52-0 hammering at Harlequins.
The hosts ran in seven tries at The Stoop, with Marland Yarde touching down twice and fellow England internationals Mike Brown and Danny Care also crossing the whitewash.
Beaten visitors Welsh have now conceded 249 points in five matches since gaining promotion from the Championship.
At the opposite end of the table, defending champions Northampton Saints maintained their position at the summit courtesy of a scrappy 19-12 success at London Irish.
The visitors led 10-0 at the interval after Luther Burrell's try and Stephen Myler's penalty.
Although Shane Geraghty was successful from the tee four times in the second half, further kicks from Myler and Lee Dickson secured the win for Saints.
Bath remain a point behind Northampton in second after they overcame last year's runners-up Saracens 21-11 at The Recreation Ground.
Jonathan Joseph's try and two penalties from George Ford saw the home side head into the break 13-6 to the good, and Kyle Eastmond went over in the second period to extend that advantage.
Chris Wyles went over for Saracens midway through the second half, but they were unable to claw back the deficit as they fell to a first defeat of the season, while Bath tasted victory over their visitors for the first time since 2010.
Leicester Tigers went down to a third consecutive league defeat as they were soundly beaten 33-16 at Gloucester, who produced a stunning first-half display.
Nick Wood, Charlie Sharples and Jonny May all scored in the opening 40 minutes and, with Greig Laidlaw successfully converting all three and adding a trio of penalties, the home side reached half-time 30-9 up.
Laidlaw added another kick in the second period to take his personal tally to 18 points, while David Mele's last-minute try was barely any consolation for a sorry Leicester.
Elsewhere, Sale Sharks defeated London Wasps 25-14 to earn their second win of the season and move above their opponents into eighth place.