Specifically for full-screen windows, grab the file with the mouse, and without letting go of it, use Command+Tab to switch to the application you want to drop it onto. Once the application window appears, you can drop the file where you want. The reason it has to be done this way is full-screen applications are given their own desktop ("Space") that doesn't interact the same way as windows in all other Spaces do.
If the window isn't full-screen, it's easier as detailed below.
For a minimized application window, drag it onto that window's icon (which will be on the right side of the Dock near the trash) and continue holding for a couple of seconds while the window un-minimizes itself. For Chrome specifically, you can then drag it into the address bar or some target on the page.
For an icon on the left side of the Dock, do the same thing except when you hold the file over the application's icon, it will display all windows from that application (including those on other Spaces). Drag the file over the appropriate window, again holding for a couple seconds, and that window will come to the front.
You can show all open windows on the currently visible Space(s) by pressing F3 (if you have the System Preferences > Keyboard > "Use all F1, F2, etc. keys as standard function keys" setting enabled, you'll have to press Fn+F3) and do the same thing as you would above with the individual windows.
If the file is on the desktop itself, it's actually even easier: press F11 (or Fn+F11 as noted above), then grab the icon, and while still holding it, press F11 again. Whatever window(s) was/were previously present will again appear, and you can drop the file where you want it.