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Java (JDK), 98 bytes

l->l.sort((a,b)->{int r=0,i=58;for(;r==0&i-->48;)r=(b.indexOf(i)>>9)-(a.indexOf(i)>>9);return r;})

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Explanation

l-> // Consumer<List<String>>
 l.sort( // Use the incorporated sort method which uses a...
 (a,b)->{ // Comparator of Strings
 int r=0, // define r as the result, initiated to 0
 i=58; // i as the codepoint to test for.
 for(;r==0&i-->48;) // for each digit codepoint from '9' to '0',
 // and while no difference was found.
 r= // set r as the difference between
 (b.indexOf(i)>>9)- // was the digit found in b? then 0 else 1 using the bit-shift operator
 (a.indexOf(i)>>9); // and was the digit found in a? then 0 else 1.
 return r; // return the comparison result.
 }
 )

Note:

I needed a way to map numbers to either 0/1 or 0/-1.

indexOf has the nice property that it's consistently returning -1 for characters not found. -1 right-shifted by any number is always -1. Any positive number right-shifted by a big enough number will always produce 0.

So here we are:

input input.indexOf('9') input.indexOf('9')>>9
"999" 0 0
"111119" 5 0
"123456" -1 -1