How can I run old games on Windows 7 with a DirectX 9 error?

I have installed Command & Conquer Tiberium Wars on my Windows 7 laptop and updates. However, when I try to run it I see this error¹:

To answer the points in the dialog:

• I have DirectX 11 installed
• The video card (and laptop) are pretty new
• I can't verify hardware acceleration; the option is greyed out (and the graphics card own menu does not appear to have this option)

I've already tried Googling this and found lots of pages of people reporting the same error but have not found a working solution. I've also tried running it with various compatibility versions and screen resolutions.

I have also tried:

• installing standalone DirectX 9
• disabling user account control (see comment below) My 64 bit Windows 7 version has service pack 1 installed.

I have also tried this on a different laptop (but with Windows 7) and got the same result.

I also saw exactly the same error with Command & Conquer Red Alert 3.

_{1 Adding text to make this more "Googleable": "DirectX Error Please make sure you have DirectX 9.0 or higher installed. Also verify that your video card meets the minimum requirements, and that you do not have hardware acceleration disabled in the Display control panel."}