Welsh claimed a 16-9 win after winger Nick Scott ran 70 metres to touch down in the dying moments of the match.
Scott picked off Stephen Donald's telegraphed pass and had sufficient pace to hold off the cover for the score, which Gordon Ross converted.
The win was their third in the Aviva Premiership and Jones was delighted his side put recent disappointments against Wasps and Saracens behind them.
"I can't quite believe it. We snatched victory from the jaws of defeat," Jones said.
"We've had a few hard lines in the last few weeks, a tough last-minute call against Saracens and we carelessly lost a bonus point last week against Wasps.
"We stuck in there and kept battling. Our effort was very good and our first-half performance was our best of the season."
Bath director of rugby Gary Gold was similarly astonished with the twist at the end and vented his anger at Donald's risky pass.
"I'm astonished, it was a rush of blood and a ridiculous pass," Gold said. "We had worked so hard to get into that position as for 79 minutes we had been effective at holding onto the ball.
"We should have retained possession and, if needs be, go back into the pocket and drop a goal."
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