How to direct connect 1-1/4 tailpiece to 1-3/8 vessel sink drain threads
You need a tailpiece extension.

Reattach the original tailpiece that came with the drain and fit an extension onto that. Extensions are available in metals or plastic and can be trimmed as needed, etc. but, unless you like clogged plumbing, don't buy the flexible "accordion-style" ones. 

Installed, it looks something like this:

(Note: This is some kind of "before-and-after" picture.) What we're talking about is the portion in the center that shows the extension between the trap and the original tailpiece (chrome).