I've run into the similar problem ever since iOS 7.0 but it was intermittent. It was also the first time I realized that I much preferred the old way of answering the phone (i.e. no swiping!).
I found that I had to go all the way from one edge of the screen all the way to other (slowly) in order to get it to answer. But try doing that when attempting to answer the call with one hand!
When version 7.02 came out, it seemed to get worse. Someone suggested that it must be a bad screen. I dunno about that. I can play games, unlock the screen, and other swiping gestures.
My iPhone is just the 5, I have a screen protector on it but like everyone else, it has never posed a problem.
I have already sent Apple a feedback asking them to please give us the ability to disable the swipe to answer so that it would act like it did before 7.xx came out.
BTW, here is the link to send your feedback: http://www.apple.com/feedback/iphone.html
Message was edited by: rob_kuhn