Warranty expires after service request, before warehouse receives laptop

So the subject just about sums up the crux of the situation:  I submitted a service request under the terms of my warranty before the warranty was up, then shipped the item.  I recently received an automated call suggesting that my laptop was no longer covered and the repairs would be at my expense.  I called Dell back and asked if, since I had submitted the request before the expiration date of my warranty support, if it was in fact covered.  They stated that the warranty would only be valid if the warehouse itself had actually received the laptop before its expiration date. 

At the time I made the request, Dell was aware that my warranty would expire soon.  I was told that the repairs would be covered in an online conversation, and this business where the warehouse must receive the laptop for repairs before the expiration date was not related at all.  I would like to know whether and how I can furnish proof that Dell is supposed to repair my laptop.

My intuition tells me that the warranty ought to cover any covered problem that arises during its validity, so long as Dell is knowledgeable that the problem occurred during a time when the laptop was covered.