One way is to keep a flag variable that keeps track of your state:

use warnings;
use strict;

my $in_section=0;
while (my $line = <>) {
    chomp $line;
    if ( $line =~ /^Success.*Id-9952895b108957daeb296ba4/i ) {
        print "$line\n";
        $in_section = 1;
    }
    elsif ( $line =~/^Success/ ) {
        $in_section = 0;
    }
    elsif ($in_section) {
        print "$line\n";
    }
}
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Update now that you've formatted your post with <code> tags: Do these Success sections always have blank lines in between them, with no blank lines in the sections themselves?