Comma

As Tolly said, there are two schools of thought about introductory adverbial phrases. As a matter of personal preference, I always use a comma following an introductory adverbial phrase -- "When he entered the room, he saw the vase broken." However, contrary to Smsretiree's answer, a comma is not required.

Tony P. asked a similar question yesterday. In answering Tony's question, we looked to two major American English style manuals for guidance.

Here is what the Chicago Manual of Style has to say: An introductory adverbial phrase is often set off by a comma but need not be unless misreading is likely. Shorter adverbial phrases are less likely to merit a comma than longer ones.

The MLA Style Manual just says: Use a comma after a long introductory phrase.  However, MLA does not attempt to define what is long.

Because the style manual guidance is a bit ambiguous, I tend to err on the side of caution, but it really is your call as the author/editor.  

When he entered the room is an introductory adverb clause and should be set off with a comma. When the adverb clause is at the end of the sentence, no comma is needed.

On a side note:

". . . he saw the vase broken"

As it stands, this phrase means he witnessed the breakage occurring. If the intended the meaning of the sentence is that he saw pieces that used to be an intact vase, the phrase should read:

". . . he saw the broken vase"