Everyone in this topic has given great advice. A few things that might help:
You need some sort of scouting, either with an observer, phoenix, or hallucination (If you are missing that many warp ins, the cost of hallucination isn't going to make or break your army). Look for an especially early 2nd gas. It means a very tech heavy play and fast lair. Double points if you don't see a roach warren. Since most roach hydra players start roach and only move into hyrdas once they have a meat shield to protect them, you can assume that they are going mutas.

Once you know he is going mutas, you need to get your upgrades going ASAP. Its your call if you want to get +1 attack so your zealots can 2 shot lings, or to get +1 armor. I prefer armor because of how much it reduces the muta's bouncing damage and speedling's fast attacks. A muta attacking a group of marines deals 13 damage/attack. A muta attacking a group of +1 armor stalkers deals just 8 damage. Throw in guardian shield, and they are down to just 5 damage.

Since you scouted his spire, (you did get your observer or hallucinated phoenix right?) you've got a choice to make. You can attack just as the spire finishes, with zealots, sentries, and at least 7-8 stalkers, and force him to waste his first mutas trying to kill of your army at his base. Mutas are pretty terrible strait up fighting units unless they are in a huge group. He should have an army of only speedlings, with 3-6 mutas about to pop. Just keep your stalkers close to your zealots, be careful with your forcefields, and you'll have completely shut down his first wave of harass. By comparison, losing those 5 mutas is like if you lost your first 2 colossi right before you were going to push in.

Ideally, you'll have your twilight council up, working on blink, 5-6 gateways, your forge working on, or finishing your 2nd upgrade, and a place picked out for your 3rd. If the zerg missed a few transfuse larva, or didn't keep an eye on his mini map, you could just win with this attack. Otherwise, you'll be well ahead. Throw down your 3rd, get up a templar archive and throw down an extra cannon at each nexus. You're now equal or ahead in ups, ahead in tech, equal or ahead in bases, and as long as you don't just let him sit and macro for 5 minutes, you'll stay ahead in army.

Your other option is to throw down two stargates and beat him in the air. Frankly, this isn't my forte, so I'll just say that it is an option that's out there.