For your code as it is.
To save typing you can define your bag array with range; You don't need to use the $i variable at all
Use $#bag as your argument to rand instead to generate a random number between 0 and the last index of the array
. You also don't need to change the value of $x at all.
This would give you

#!/usr/bin/perl -w

use strict;

my @my_bag;

@my_bag = (1..10);

while (@my_bag > 0) {
        my $x = int(rand($#my_bag));
        print splice(@my_bag, $x, 1) . "\n";
}
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Shuffling via rand and splice isn't very efficient though on large arrays. Have a look at perldoc -q shuffle for an example of a fisher-yates shuffle