Price match policies vary wildly by hotel and booking platform, so your actual leverage depends on where you booked. If you reserved through their direct website, you're in the strongest position - most chains will adjust the rate or let you cancel and rebook at the lower price with no penalty. If you booked through a third-party site like Expedia or Booking.com, the hotel has less incentive to help since they didn't lose the direct booking relationship.
Here's the practical angle: before calling to ask for a match, check if you can simply cancel and rebook at the new rate yourself through the same channel. Many hotels offer free cancellation up until a certain date, and if that applies to you, it's the fastest solution that doesn't require negotiation. If cancellation isn't free or has already locked in, then contact the hotel's front desk directly rather than going through the booking site - they have more authority to adjust rates than customer service lines do. Mention you're a repeat customer or planning to spend money there during your stay; it gives them a reason to say yes beyond just matching a competitor's price.