How to keep thick puréed sauces from separating into solids and liquids?
SAJ14SAJ has mentioned starch, but I wouldn't add any additional -- I'd just finish the pasta in the sauce.

Pull your pasta a minute or two early, let drain but do not rinse it, and add it to your sauce.  The pasta will finish cooking in the sauce, absorbing some of the liquid.  It will also release some of its starch into the liquid, helping to bind it.

If you can, reserve some of the pasta water to thin back out the sauce in case thickens too far.  (A pyrex measuring cup works well for this, as you can just dip it into the hot water before you drain your pasta)