Yes and no. Depends on the sensor.
Many of the recent dSLR models have an "ISO free" sensor, i.e. it doesn't matter whether you expose correctly at a high ISO setting, or underexpose at the native sensitivity. AFAIK, not all have, and older dSLRs don't have that kind sensors. For a non-"ISO free" sensor, pushing an underexposed shot in post will give more noise than upping the ISO and exposing correctly. With my dSLR, which doesn't have an "ISO free" sensor, I like to "expose to the right" (ETTR), i.e. overexposing slightly - while, of course, making sure not to blow out the highlights - and pull the image in post. That makes for a slight improvement in noise, particularly in the shadows. See
Digital Exposure Techniques for some info on the ETTR trick.
And no matter your ISO setting preferences, personally I'd rather have a somewhat noisy, but sharp image, than an image with motion blur or unacceptably low DOF. If I can accept blur, I might as well add it in post, by using heavy noise reduction