Centre Tom Cheeseman scored three tries as Meehan's men dug deep for victory but they were forced to withstand relentless pressure during injury-time before claiming a win that keeps them second in the table, three points behind leaders Gloucester.
Meehan said: "Worcester are a good side, so this was always going to be a difficult game.
"We are delighted with the win - it is a massive result for us and has provided good momentum for our next Premiership match against Saracens.
"The defensive effort was fantastic. Those last few minutes were all about how big the players' hearts were, and it took a huge disciplined effort."
Bath could end up meeting Worcester in next month's European Challenge Cup final, and on Saturday's evidence it would be an occasion to grace any competition.
The lead changed hands four times, with Cheeseman's treble keeping Bath afloat despite the best efforts of Worcester wing prospect Miles Benjamin.
His performance set the tone for a spirited Worcester recovery from 16-8 adrift, before Cheeseman administered the killer blow.
Fly-half Shane Drahm added a third Worcester try, while he also booted a penalty and conversion, but the Warriors had to be content with a losing bonus point.
They also lost skipper Pat Sanderson to a first-half leg injury that could make him a doubt for next Friday night's Challenge Cup semi-final against Sixways visitors Newcastle.
Meehan added: "This is a very good Worcester side.
"I think they will build on this season and be pressing further and further up the table next term."
Worcester rugby director Mike Ruddock had no complaints about the result after Warriors suffered their first home defeat for two months.
Ruddock said: "We made crucial errors far too often, and our skills weren't as good as Bath's over the 80 minutes.
"We lost the game, and that hurts. We felt we should have won after getting 20-16 ahead, and we have got to learn to shut the gate and stop teams coming back at us.
"Bath are a quality team and I like the way they play with their width and tempo."
Ruddock also saluted Benjamin's contribution, adding: "We have looked to develop young guys like him at the club.
"He looks a real candidate for England honours in the future.
"Teams are trying to take away our set-piece game in matches and work us out. We have needed a plan B, and guys like Miles, Rico Gear and Marcel Garvey give us that."