On another day Welsh's four-try haul might have been enough to see them snatch a victory, and the Exiles did lead 10-9 after 33 minutes thanks to Dan George's converted try, only for the home side to blitz the visitors either side of half time.
Tries from Kyle Eastmond and Michael Claassens saw Bath turn that one-point deficit into a 21-10 lead, and when David Wilson and Will Tanner crossed early in the second half Bath were in control.
But in a game punctuated by substitutions, the Exiles finished the stronger. Carl Fearns may have cancelled out Jonah Holmes' try for Welsh, but Paulica Ion and Ashley Smith ensured a more than a respectable look to the final scroreline.
Tom Arscott's withdrawal from the original line up ensured Welsh took the field with just one of the club's summer signings on initial parade, Nick Runciman.
Tom Heathcote's penalty was cancelled out by the boot of Ross, whose ninth minute strike registered London Welsh's first points as a Premiership club.
Heathcote restored the home side's lead, albeit with the assistance of the post, after Welsh were penalised at a scrum. The first half was descending into a cat and mouse kicking duel.
Jack Cuthbert's grubber teased the hard chasing Eastmond but with play brought back for a Welsh infringement Heathcote increased the lead.
However, seven minutes before the break Welsh stunned the home side when Bath's lineout malfunctioned and Runciman didn't need a second invitation to counter.
Welsh worked the ball through the hands of James Lewis and Joe Ajuwa, with former Bath prop Billy Moss getting also taking it on. Bath infringed but rather than take the three, Ross pushed the ball into the corner and Exiles got their reward when George broke off the maul to dive over. Ross converted.
That try, though, only served to sting the home side into life and a lovely offload from Ollie Devoto sent Eastmond bursting through to score. A minute later Eastmond looked in again, but for a try saving tackle from Nick Scott. Michael Claassens, though, went over from close range.
Bath began the second half as they'd finished the first. The pressure told when Wilson took a flat pass and there was no stopping him from just a few metres out. Devoto added his second conversion and the Exiles now trailed 28-10.
That scoreline made even worse viewing for those London Welsh supporters among the crowd when Devoto prodded a penalty into the corner and his pack did the rest, with Will Tanner the last to emerge to take the plaudits. Devoto's conversion left Welsh 23 points adrift.
But Welsh would reply with arguably the try of the game, a superb length of the field effort. Seb Jewell's miss pass released Phil MacKenzie and although the Canadian international was collared, Welsh kept the ball alive and Jonah Holmes finished off the move by the posts. Gavin Henson converted for his first points in Welsh colours.
As if angered by Welsh's impertinence, Bath's riposte would be swift. Devoto put another penalty in the corner and Fearns burst clean through several tackles at pace to score.
Devoto's conversion took Bath's tally to 40 points, but although 24 minutes still remained it was Welsh that monopolised the scoring. Two Henson penalties allowed the Exiles to work their way up field and although Alfie Toa'ala was adjudged short, Welsh were not to be denied and Paulica Ion forced his way over against his former club.
Henson bisected the uprights for the extra two points and then produced a fine touchline conversion to close out the scoring, after Matt Keyte's quickly taken penalty resulted in Ashley Smith crossing in the corner.
Bath Rugby: Tries: Eastmond, Wilson, Claassens, Fearns; Cons: Devoto (3); Pens: Heathcote (3).
London Welsh: Tries: George, Holmes, Ion, Smith; Cons: Ross, Henson (3); Pens: Ross