The famous problem with the N1 was plumbing all of those engines to the fuel tanks without leaks. But SpaceX uses a similar design (lots of small engines) - how have they solved the plumbing problem?
Control systems and manufacturing and simulation were already mentioned, but a more critical issue is that N1, due to various design issues, had no way to do static tests that other rockets went through, meaning many issues were found when the rockets underwent Rapid Unplanned Disassembly in flight or on pad.
SpaceX built a much smaller, simpler rocket - and had a way to do the testing before flying.
I think a lot of the answer is just really good engineering. Precision manufacturing and computer simulation have come a long way in the past 50 years.
It's more accurate to ascribe it to better manufacturing processes, really. They're the result of engineering, but orthogonal to what you'd think of as engineering practice.
I wouldn't say that plumbing was the biggest issue. A big limiting factor was the control systems available then. I believe a couple of the N1 failures were caused by control system failures. With modern controls you can monitor and control each engine with much more granularity.
SpaceX built a much smaller, simpler rocket - and had a way to do the testing before flying.