When your client escalates an issue to your organization, it can be very stressful, but it can also be a chance to shine as an organization if the escalation is handled quickly and professionally. There is no way an organization of people can always be perfect, so the question really becomes is your organization one that can learn from its mistakes, handle others mistakes professionally, and focuses on solution vs. blame.
One of the most important ways to shine is to respond promptly to client escalations. This doesn’t mean you have to have a solution immediately. It just means you quickly respond stating you understand the client’s concern and are looking into the problem. The last thing a client wants to have happen is to escalate an issue and not hear anything back. As your investigation proceeds, ensure your client knows when and how you will give them their next update on the investigation. At the designated time, always send out an update, even if no new information is known (never go radio silent).
Throughout your correspondence, you should avoid jumping to any conclusions (especially around whose fault it is). Assigning blame or falling on your sword early might make you cognitively feel better but it can hinder the investigation (everyone more focused on protecting themselves vs. solving the problem) and it can make your company look incompetent if you later have to admit the early judgement was wrong. Relay facts as you know them and explain how you are investigating them but do not make judgements until you are certain.
Engineers especially have to be assured that you are looking for a professional response not blame. At this time you want the best of their intellectual talent being spent on providing a professional analysis with good reports containing facts that can be reported back to the client. Emphasize that this is a chance to learn and make improvements for the future -- focusing on a professional response is the best way to help the client and the company.
Be aware of the political nature of escalations in an organization. The higher up in an organization an escalation has to go to get resolved, the more churn it creates (most of which is people trying to avoid getting blamed). Always try to resolve things at the lowest level possible, if the issue can be handled by front line managers, handle it there. If it can’t be then ensure that every group involved had someone review and give input before notifying the next level of management. The last thing you want to do is to have an issue get escalated up in one branch then have it come down from above on another branch to a team who was unaware of the issue -- this can lead to higher chance of misinterpretation, more people needing to be kept informed, and a higher desire to protect teams from executive judgement. You also don’t want different branches of the organization disagreeing about “the facts”.
Finally make sure you conclude the client escalation definitively. Be sure that your client understands what the issue really was, what the impacts were, what steps were taken to correct the issue, and what steps will be taken in the future to avoid a similar issue in the future. If your organization was at fault, then have the most appropriate management personnel acknowledge that fault.
A client escalation can actually give a client with more confidence if you handle the escalation with prompt, accurate communication of the facts and have a professional resolution.