How can I take browser screenshots at a higher resolution than my browser supports?

You can do it with Firefox and 2 extensions: Web Developer and FireShot.

Once both extensions are installed, go to Tools - Web Developer - Resize - Edit Resize Dimensions... .

Add a new size, 4000 x 3000. check "Resize the viewport" if you want only the page content to be 4000x3000. If you don't check it, the complete window of Firefox (with toolbars, menu, ...) will be set to this dimension.

Once at the correct size, go to Tools - FireShot - Capture Entire Page and ... . Select an action, like Save for instance. It will save the page content that has been set by Web Developer to the desired size.

Meanwhile (for at least a year or two -- since version 15, if I am not mistaken), Firefox supports this directly via the integrated developer tools.

Either hit CtrlShift M or select Responsive Design View from the Developer Tools icon in the stripe menu.

That will bring up this view, which will let you choose from a set of presets as well as enter any desired resolution and directly save a screenshot in PNG format to disk with one click:

On a Mac, Paparazzi can help, either using its GUI or using a URL syntax like:

paparazzi:(width=4000,height=3000,maxheight=3000)How can I take browser screenshots at a higher resolution than my browser supports?

The URL syntax can easily be turned into a bookmarklet. Paparazzi uses the WebKit rendering engine.