That's not true. CPU usage is the same, but it requires more from your GPU to run at a higher resolution.
Let's say for instance, that you run a 1366x768 screen, your CPU can deliver 40fps, and your GPU can deliver 60fps. You will only get 40fps, because your CPU can't feed your GPU quickly enough.
If you upgrade to a 1920x1080 screen, your GPU may only be able to deliver 35fps at that resolution. You're still running the same game, and your CPU can still deliver 40fps, so it's no longer the bottleneck, but you can't magically get more CPU performance by increasing your GPU workload.