The first time a client escalates directly to you, the instinct is to solve the actual problem immediately. Resist that. The first job isn't fixing it, it's making the client feel heard and in control before you've fixed anything, because an unacknowledged escalation gets angrier by the hour, and a rushed, half-right answer usually makes it worse, not better.
Acknowledge fast, ideally within a couple of hours, even if all you have is "I'm on it." Don't promise a fix or a timeline you haven't actually confirmed internally, an over-promise you can't keep costs more trust than the original problem did. Loop in whoever needs to know, your manager, the account owner, before it becomes a surprise for them too. And follow up even when there's no news yet, silence is what turns an escalation into a bigger one.
Three moments, three short messages. Adjust the specifics, keep the shape.
"Thanks for flagging this, and I'm sorry for the disruption it's causing. I'm looking into it now and will have a real update for you by [specific time], not just a status check-in. If anything changes before then, I'll reach out sooner."
"Here's where things stand: [what actually happened, briefly]. Here's what we're doing about it: [the fix, in progress]. Here's what this means for you specifically: [impact, or lack of one]. I'll follow up by [specific time] whether or not it's fully resolved."
"[Client] escalated [issue] to me directly at [time]. I've acknowledged it and told them I'll follow up by [time]. Wanted you looped in before they mention it to you, here's what I know so far: [summary]. Let me know if you want to be on the next update."