The warranty won't cover a fried CPU. However, CPU's have internal circuitry that shuts them down before bad things happen.
Just take the system apart.....take out the CPU and clean it with dab of isoprophly alcohol on a clean piece of non shedding cloth / chamois. Apply the new paste as per manufacturer's instructions.
Don't scrimp on the cooler. A Scyth Muegn 2 Rev B SCMG-2100 would be my recommendation at $40 from newegg.
Don't try and do this within the case. Take the MoBo out and do it on a clean surface, preferably the original MoBo sleeve it was shipped in.