Leeds boss Marcelo Bielsa could not explain how his side threw away a three-goal lead to draw 3-3 against Cardiff.
Bamford added a second from the penalty spot early in the second half before Cardiff clawed back to level terms with goals from Lee Tomlin, Sean Morrison – who was sent off shortly after – and Robert Glatzel.
"I did not get the preparation right because I knew about Cardiff City's strengths and we didn't resolve it," Bielsa said.
"We knew how dangerous they are in the air and from set-pieces. Set pieces are crucial in this division and Cardiff made us pay in that area today.
"Some of our attacking play was the best I have seen from Leeds and It is difficult to explain the result.
"There is no way we should have drawn that game. l felt the substitutes were OK, we needed to take Bamford off to freshen things up a bit.
"We wanted to give Stuart Dallas more help down the left and to get more height out on the pitch.
"I don't look back on the subs and think I got it wrong. I felt that we were defending OK and even at 3-2 the team was calm.
"Defensively we were not that bad, but they have exploited us for the third goal."
But delighted City boss Neil Harris hailed his side's ability to hang in there.
"Leeds are a great side, the energy they have, the ability they have and the forward runs they have make them a great side," he said.
"They are as one of the best sides I have seen in the Championship in years.
"They were better than us, but we gave them a leg up, we gave poor goals away and their penalty was debatable.
"The boys have shown great character to come from behind, but we cannot give good teams a start and shoot ourselves in the foot.
"We got off to a bad start, Leeds are very dangerous on the counter-attack and we did not box off the edge of the box as well as we should have done.
"Lee Tomlin's goal was a brilliant finish, he has got great ability and that moment got us back into it.
"I have got 11 points from my six games in charge so I am really pleased with that return.
"The message is getting across and the confidence is growing."