Actually the proper sentence is: This pan is the same AS THE ONE THAT I bought yesterday. When you use "same as", it should be followed by a noun phrase (This pan is the same as THAT PAN.)or a pronoun (My bag is the same as yours.) that you are comparing the subject to.

The missing phrase is 'as the one' "This pan is the same as the one I bought yesterday." Just saying 'as' is correct, but a little less clear to the listener. The listener will ask themself - 'Same as what ??" and use the context of the preceding conversation to understand. I would always use 'as the one' myself to make it clearer.