There are various things going on here.
Firstly a DJ performing with legitimate media, (i.e. not crappy mp3s), will enjoy the benefits of mastered material to work with.
The individual parts that make up their track will also have gone through a multitude of processors.
This makes DJs tough acts to follow.

There are ways & means though.
Firstly gain structure is sooooo important. How did you connect to the PA? We’re you supplied with quality active Di Boxes like KT or BSS?
Second, I take it you ran into a console? Who was at the helm & how knowledgable were they? Bit of compression on the console & an attentive pair of ears will make all the difference!
Thirdly, when prepping material for live performance, check it thru a spectrum analyser one track, (instrument), at a time & try to find little pockets for each within the freq spectrum. Also take particular note of the bottom end as there’s a far a chance they’ll be more going on than your studio monitors can reveal! Massive bass bins in a club environment can provide quite a surprise lol