A Tim Visser try and the boot of Greig Laidlaw gave the Scottish outfit a 19-3 half-time lead but Baber's troops fought back after the interval.
Dan Parks kicked two penalties and Leigh Halfpenny grabbed a late kick himself to leave the final score at 19-12 in Edinburgh's favour.
The losing bonus point keeps Cardiff on top of Pool Two, now level on 13 points with Edinburgh, and Baber insists his pack should get all the applause.
"It seemed like Edinburgh were getting the ball they were getting the three's and the sixes and that was causing us all sorts of pressure," he said.
"And equally so, when we were getting the ball we didn't hold on to it long enough and when you give possession to a team like Edinburgh they are going to cause us problems and that is exactly where we felt the pressure going into half-time.
"Undoubtedly you could see that they were hurting from last week and they came out of the blocks quite early against us and they managed to get some points on the board and that creates momentum and as a mind set it takes something to get back into it.
"Fortunately I thought we were good in the second half and praise must go to our forwards.
"I thought they battled hard to get us in there, created two or three opportunities that we probably should have finished as well and fortunately we come away with a point.
"The physicality of our forwards started to create opportunities for us and got us going forward and created some space for us to go and play and eventually I thought we could have got a couple of tries out of it.
"I think we were unlucky with a couple of stumbles but it just doesn't come off for you sometimes."
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