The famous baby on the steps scene in THE UNTOUCHABLES was a late sequence put in because they didn't have the money for the elaborate train chase scene that was originally scripted by Mamet.

So since they were out of time and money, and Mamet couldn't be bothered (he was directing something by then) De Palma put in an homage to Battleship Potemkin, cheaper because it's one location, and designed the sequence himself.

It became the centerpiece of the film.

I wrote about it here: http://writerjoshuajames.com/dailydojo/?p=999

The first one that comes to mind is the infamous scene in Raiders of the Lost Ark where Indy shoots the swordsman rather than get into a protracted fight with him.

The way Spielberg likes to tell the story, Harrison Ford was quite ill that day and said he had maybe an hour of shooting in him. Speilberg said, "Harrison, the only way I could do this in an hour is if Indy just shot the guy." That got a laugh from the crew, and so the change was made on the spot.

Speaking of Spielberg, this wasn't due to lack of money, but the whole first two thirds or so of JAWS, where the shark is mostly unseen is so effective because the mechanical shark built for the movie malfunctioned so much that Spielberg was forced to shoot around it. Or maybe it simply wasn't ready for a long time, I can't remember specifically.

The ultimate example of this has to be MONTY PYTHON AND THE HOLY GRAIL. They didn't have money for horses, hordes of extras and obviously judging from the solutions they came up with the movie is infinitely better and funnier for it.

As for ROCKY, I'm ashamed to say that even though I'm a child of the 70s/early 80s and have seen bits and pieces of it, I've yet to see the whole thing from start to finish. I'll have to rectify that soon.