The Knicks had led by 17 points after the first half, but Jarrett Jack's season-high 31 points, along with 10 assists and five rebounds, helped to push the Cavs to victory.
"I thought we started out well," Woodson told reporters after a tense encounter at Madison Square Garden.
"We came in after the halftime, I thought we were playing pretty good basketball, then we went dry in the third. We came out in the third so slow.
"This was like a getaway game to get us set for the road and didn't close it out. We just didn't handle our business and I have to take responsibility for that"
The loss sees the Knicks occupying third spot in the Atlantic Division, with a record of 29-41 for the season.