But the minister for education, lifelong learning and skills also said there were two sides to the story, with some poverty-pleading schools sitting on bumper cash reserves.
Speaking at an NUT Cymru fringe meeting of Welsh Labour's annual conference in Llandudno last week, Ms Davidson answered frontline critics who warned of redundancies and school closures.
"We said you should only pass the cut on to frontline services, like schools, if you know they have the capacity to make savings," she said.
"But some local authorities passed them on automatically which meant they put some schools in deficit."
Assembly government figures out last December show that cash reserves in schools fell by pound;12 per pupil in 20056, but still add up to pound;72 million. However, some education unions argue that the figures are flawed because they do not account for all money going out over the financial year, and big reserves are often indicative of big projects, such as a new ICT suite.
The government expects all authorities to make a 1 per cent saving until 2008 as part of a drive to save pound;600m by 2010 across all public bodies.
In January, Ms Davidson wrote to LAs warning them not to pass more than 0.3 per cent on to schools. But Neil Foden, head of Ysgol Friars in Bangor and a member of the NUT national executive, said: "There is a huge amount of resentment about efficiency savings because they are cuts. In too many authorities the 1 per cent has been passed on."
He said teachers were struggling with endless new initiatives, many of which did not stack up.
"Collaboration is a mantra rather than something we can evaluate," said Mr Foden, who joked: "Since the last Assembly election I've probably had more partners than Liz Taylor."
He said geographic pathfinders, which review education in specific areas, were skewed in favour of the tertiary sector, and the pound;16m Raise fund for disadvantaged pupils had not given schools what they expected. Workload was still a problem and the advisory panel should be sharpened up: "Give it teeth or get rid," he said.
Ms Davidson, who made a point of describing Labour as a "socialist party", restated her commitment to children in Wales's poorest areas.
She said investment was needed in the early years and better access to vocational qualifications. More than a third of post-16 schools and colleges will be offering the Welsh baccalaureate by September.
And she insisted that spending per head on education in Wales was higher than in England: pound;712 compared with pound;698.