Congrats on the new job ! Believe me, taking a counter-offer is almost always a bad idea. If you have a reasonable request (ie: be paid fair market value for that role, not 2x market or whatever), then take it to the boss before you give notice, and see what they say. If they arent interested in that, then move on if you can.
If they try and counter only after you give notice, keep in mind that they only were interested in entertaining a valid request once you held a gun to their head.
So, if you take the counter, then that establishes an unhealthy negotiating precedent. After that, the next time you have another reasonable request about something, the only way you'll get them to listen seriously is to hold a gun to their head again by threatening to leave. If they are unwilling to talk without the threat of you leaving, then that speaks volumes of how much respect they have for you. They may not have the ability to pay the $ you want, but if they wont talk about what other perks or whatnot they can offer, then it's time to go.