Open your image in Photoshop.
First make sure the image is in RGB mode.
Go to: Image > Mode > RGB color
Then make the layer independent by double-clicking it in the layers window.
With one of the selection tools, select a portion of the background color.
Go to: Select > Similar
And all of the background pixels are selected.
Hit delete on the keyboard and you have a nice clean logo with perfect edges.
If the text in the logo is the same color as the background, go in quickmask-mode and paint in the text.