I have discussed the phrase “on the day” — the American equivalent being “on the day of [the event] — a couple of times, most recently here. But I had never encountered “on the night” until a couple of days ago, when I read, in the Philadelphia Inquirer, a quote from Jim Curtin, the manager of the Philadelphia Union soccer/football team. Referring to the Union’s close defeat in an important game, he said, “Our players played with an intensity that I think made the fans proud on the night.”
I didn’t know Mr. Curtin’s nationality, and when I looked it up I found he came from Oreland, Pennsylvania, a Philadelphia suburb not fifteen miles from where I’m sitting now.
notoneoffbritishisms.com 
I believe in the theatre the phrase “It’ll alright on the night” means we’ll iron out our mistakes. It was also the title of a ong-running bloopers compilation TV show.
It’s commonly used in Australia – ‘she’ll [or it’ll] be right on the night’ means, essentially, that at the pointy end, everything will come together and it will be fine. I agree with Paul, it’s probably originally a theatre expression that’s now so widespread we’ve forgotten where it came from.
BTW, lots of things are referred to as ‘she’ here.
There is an unspoken “in question” following either “on the day” or “on the night”. It’s a quick way of specifying a particular time. The full phrase is probably only heard in court nowadays.
There is a popular British TV programme of funny outtakes and bloopers called ‘It’ll Be Alright On The Night’.