First, get the facts; who what, where and when.
Collect any error messages and diagnostic data.
Record a recreation scenario.
Don’t forget the business impact! It is more significant that a user or client can’t transact than is a feature is not acting as expected. This will drive both the urgency and scope of the escalation.
Store all this info in a cool, dry data repository for sharing later.
Reach out to your management and describe the incident. Get their direction on who to contact next. This may be their Acct Manager, CS rep, Product Manager or a Development Manager. Business impact will drive who to get involved.
Share the documents. Be prepared to answer several questions from internal folks such as what settings are active, any log data, and several what-if scenarios.
Be the liaison to the client. This means describing the actions they can expect to address the incident and a preliminary timeline for resolution. This, of course, is all best done in writing.
Create additional data points as necessary, a product Jira ticket for example.
Schedule status meetings internally on a regular cadence and regular update meetings with the client until the issue is resolved. With the help of your manager, Support assumes the burden of working with the client to try out and verify solutions and get more diagnostic data as appropriate.
Once the issue is resolved, meet again with the internal team in order to determine what went well and how the process could be improved.