It is difficult to comment on your appraisal, because I am not familiar with your area, but I do know that appraisers are being pretty conservative in their valuations these days. If you came in at contract price, you are ahead of the game. I don't know if you are working with a Realtor or not, but if so, he or she should be able to go over the homes that were used as comparable sales with you. You need to contact your lender to see if they will provide you with a copy of the appraisal. That way you can see what homes they used and if, and how, they made the adjustments to come up with that price. Also, if you are financing the home, chances are the lender will put the appraisal through at least one appraisal review, and quite possibly more than one. They will not lend you money on a home if they feel it does not have a reasonable expectation to hold its value over the life of the loan.
As far as Zillow.com is concerned, if you will read their disclosures, they tell you that their valuations are not guaranteed. For example, I had a home listed that was a 1965 home with no garage, a second bath in a laundry room that was added into a closet, and window air, in an older section of town. When you pulled the property up on Zillow, they compared it to new homes in a brand new retirement community with a marina, a luxury clubhouse and Olympic-sized pool that overlooked a 16,000 acre lake. When I look at Zillow, I select out the homes from the ones Zillow gives that compare closely to the one I am trying to get a valuation on. They strictly go by area and square footage, with no adjustments for age, condition, lot size, etc.
Most appraisers won't appraise a home for more than contract price anymore. Also, in some cases, they can't because of lender guidelines. By not doing so, it gives them an extra measure of security, also, just in case their judgment of the value ever comes into question. If you want to get a real deal, you have to find a Seller who is willing to take less money than similar homes in the area were selling for. More than likely, if it wasn't a Buyer's market right now, you would never have gotten the home for the price you did, because the values are down. Also, if the market values weren't down, you also would not get as good of an interest rate as you are probably getting on your loan. I'm pretty sure you will probably do o.k., so don't let this discourage you.
Good luck in your new home!