The West Country outfit showed admirable spirit to grind out the win courtesy of five penalties from Nicky Little and a clinical first-half try from Michael Stephenson.
It was the perfect way to start 2010 after the drugs-related scandal which hit the club last year and Meehan felt his players deserved credit.
Defeat at Headingley would have sent Bath bottom and Meehan said: "A lot of clubs could have quite easily imploded under those kind of pressures that we were under.
"Even when we were having a poor run of results, the morale was still quite high and I don't think people can quite grasp that, or maybe they don't want to. But it was there. It was great to finish last year and and begin this one with another victory.
"We've got players to come back but the guys that are there now are fighting for the club and the supporters."
Both sides came into the match on the back of morale-boosting victories last weekend - Bath beat local rivals Gloucester convincingly at home and Leeds won 16-15 at Newcastle.
There was precious little free-flowing rugby on display but Bath ultimately had the calmness of Little and finishing power of Stephenson to thank for their win.
Leeds looked dangerous when on the attack but they could not penetrate the visitors' line and fell behind for the first time in the 29th minute when Bath claimed the only try of the game.
The score stemmed from an outstanding run down the right flank from number eight Luke Watson, who was held up just short of the line before Stephenson followed up to burrow his way over the line.
Leeds claimed five penalties from fly-half Joe Ford but are now rated as clear favourites for the drop.
They have not won at home in the GUINNESS PREMIERSHIP all season but head coach Neil Back believes there could be plenty more twists and turns between now and May.
He declared: "Can we stay up? Absolutely. The guys won't fail through lack of effort, but we're on 14 points and there are still 50 points up for grabs. While there are still points up for grabs we will maintain our confidence and belief, and why not?
"We played very well in large parts today and we know the squad is growing with every exposure in the GUINNESS PREMIERSHIP. There will be a lot more twists and turns, and a lot of points on offer, and we will keep fighting.
"I love working at Leeds Carnegie, they're a great bunch of guys to work with. We need to keep going between now and the end of the season."
