Who are you interviewing with? If its your current supervisor or someone you have worked with, then bring up the times you improved something/averted disaster. If its someone entirely new to you, then it will be tougher, but use the same examples.
Use lots of examples in your interview of how you accomplished something and how that translates to your new role. Be confident in your ability to do the new job(because if you do not think you can do it, why would someone else?). Do not jump into how you can improve things too much, since this might be a role where the interviewer already oversees the whole group, and if (s)he thinks you are impugning his/her ability to lead, then your resume will find its way into the trash bin. Talk about how you improved things from your current role, rather than what you will do in your new one.