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."
Gilbert has released a new line of rugby cleats. The Gilbert Virtuo 8S is part of the exciting new product. Check it out.
The Barbarians are one of the top invitational rugby sides with a long history and classic rugby jersey.
The Nike Tiempo is a solid rugby cleat and one of few styles still made from full-grain natural leather.
The Lions get ready for their matches with this green training jersey. It's what the players wear. Get in the Gear!
A cool looking all black rugby cleat with the high performance adidas is known for. Get in the Gear!
Wear the crest of the British and Irish Lions on your t-shirt. A great look for the summer.
The Lions are ready to get their Australia tour underway. They arrive in Perth on Monday.
The New Zealand All Blacks training jersey for 2013/14. Get in the Gear!
The USA Rugby Pro Alternate rugby jersey is perfect for any fan of the Eagles. Get yours to wear during the summer Test matches.
The NEW All Blacks 2013/14 jersey has arrived at World Rugby Shop. Dare to wear the colors of the All Blacks.