Ireland fly-half Jonathan Sexton kicked six penalties - two during the last seven minutes - to give Pool Three leaders Leinster a tense 18-13 triumph.
Bath's England wing Matt Banahan scored the only try of an attritional battle after 65 minutes, but a second defeat in three group games means the west country club have little chance of progressing to the quarter-finals.
The minimum requirement will be victory over Leinster in next Saturday's Aviva Stadium return, but Bath rugby director McGeechan said: "I am very proud of the players - their attitude was superb.
"We defended tremendously well, restricting them to penalties, and it is about the small margins.
"We made one or two errors. They got the ball into the right areas quickly and put the pressure on. They kicked the penalties.
"We probably didn't have quite that same amount of extended pressure, but we scored a very well-executed try, and that showed the attitude was really positive.
"The work-rate when we didn't have the ball was superb, and our attacking play was very encouraging. We just need to have a bit of composure at key times, but it is a relatively new team with new combinations, and that is what you grow towards.
"I think it took Leinster probably three or four years to get to that point. You don't get there overnight.
"Leinster have got huge experience and they have got good players in key positions. When you win Europeans Cups, that's what you get. You know how to do it, you've been there before, and so it grows.
"They bided their time very well, and that is experience. That is a good team with good players reading the situation well."
Centre Olly Barkley, who slotted two first-half penalties for a 6-0 lead, converted Banahan's try to edge Bath one point in front, but Sexton nudged Leinster towards a priceless away win when home skipper Francois Louw infringed.
And to complete a miserable two-minute spell for South African Louw, he was then sin-binned as Bath ended the game a man short.
Bath now head to Dublin and, while they are not mathematically out of quarter-final contention, it will require a Herculean effort to turn things around, with a losing bonus point providing scant consolation.
Leinster were guilty of wasting some clear opportunities, which made their task more difficult than it otherwise might have been, and that profligacy was not lost on their head coach Joe Schmidt.
"The frustration was that we couldn't get the ball over their line. We appeared to have a couple of gilt-edged chances," Schmidt said.
"We want to be a lot more accurate to make sure we finish opportunities that we create, but I am delighted with the result.
"Against a quality team it's hard to create opportunities, and the last thing you want to do is butcher them like we did today. We almost suffocated ourselves at times."