Logically, 差点没 is not that different from 差点. In fact, you shouldn't think of it as a single word; think of the 没 as belonging to what follows 差点 and it makes more sense. Still, it can be a little confusing, and is worth a closer look.

Basic Structure

Subject + 差点 + 没 [Verb Phrase]

Note that the "Verb Phrase" part is a desired result. So this structure is literally saying, "almost didn't [Verb Phrase]."

Also note that because you're using 没, you don't use 了 after the verb phrase.
Examples
wǒ chàdiǎn méi ɡǎn shànɡ huǒchē。
我 差点 没 赶 上 火车。
I almost didn't make the train. (but I did)

měiɡuó duì hěn yǒu shílì，wǒmen chàdiǎn méi yínɡ。
美国 队 很 有 实力，我们 差点 没 赢。
The American team is very good, we almost didn't win. (but we won)

shíjiān tài jǐn，zhèɡe xiànɡmùchàdiánér méi zuò wán。
时间 太 紧，这个 项目差点儿 没 做 完。
Time was almost up, we almost didn't finish this project. (but we did)