You have to say:

He happened to break the vase.

It would mean that he broke the vase accidentally.

Tom was dusting off some things in the living room yesterday.  He happened to break a vase.  His mother was angry.

You need to change "broke" to "break." Write "He happened to break the vase." Then it is perfectly correct and natural. "

The verb form here is the "infinitive," "to break." It is often the same as the word to + the present tense.

I break eggs when I make an omelet.

To make an omelet, you need to break eggs.

I write everything on my computer.

I find it difficult to write things in pen or pencil.

"Broke" is the past tense of "break." You use it for things that happened in the past. Some English verbs form the past tense regularly by adding the ending -ed, but many are irregular and have a special form in the past. 

I break eggs when I make an omelet. Yesterday I broke eggs for an omelet.

I write everything on my computer. Yesterday I wrote an email to my daughter.

I see sunlight coming through my window. I like to see sunlight coming through my window. Yesterday I saw nothing but dark clouds through my window.