How do I neutralize a strong garlic flavor?
For the next time you make it, a common way to take out the "bite" of raw garlic is to roast it first. Chop off the top of the unpeeled head, drizzle in olive oil, sprinkle some salt, wrap in tin foil and pop in the oven. More details and pictures covered here and here.

A quick search for roasted garlic pesto came up with a bunch of recipes as well. I've used this trick with numerous recipes when I want lots of garlic, but want to take out the bite (from salsas, to guacamole, to just a straight garlic spread for bread).