I like to exagerate whatever real emotion I'm feeling and make it a game of sorts. The emotion I most often exagerate is frustration when dd (3yo) won't go along with my requests. I cross my arms, pout and act powerless and disappointed in a very comical way that is obviously a charade and never intended to make dd feel badly about her choices. She laughs and bosses me around. Nine times out of ten she'll start telling me to do the very thing I was asking her to do. I loudly protest that I'll turn into a stalk of broccoli or some other ridiculous comeback. Once we play through the entire scenario (when she's giggling and laughing too much to continue ordering me around), I can usually guide her back to the activity she was refusing to participate in and accomplish our task.
I do realize every child is different, and this approach might not work for all kids. However, what I truly love about this approach isn't how it helps me control dd - I love how it changes my perspective. When I make a "game" out of an emotion I am really feeling (frustration, annoyance, etc), it won't be long before I get into the spirit of the game and that original emotion melts away. Also, it sounds time consuming, but we actually don't have to play this "game" very often at all. Think of it as an investment that you make when you can (when you're not in a rush, late getting out the door), or better yet, think of it as dropping coins into an emotional piggy bank. I feel that insisting the child goes along with your goals is perfectly fine and sometimes quite necessary, but remember that you are withdrawing coins from the piggy bank and you will need to pay them back before the bank runs dry. After all, being a small child, with no access to family resources, limited recourse if you feel wronged, lesser abilities to communicate or negotiate, and subject to the apparent whims of your parents is a fairly powerless kind of experience, wouldn't you say? No wonder kids try to refuse our requests; in fact, I think they need to in order to figure out how the world works and how much control they really have over themselves.
If you're not sure whether this would work in the moment, make time to play out a similar scenario when you don't have a necessary activity like handwashing to accomplish. Invite your lo over and jokingly command him to do some silly and impossible task, like put all his toys on the ceiling. When he refuses or looks at you like you're crazy, go into a gentle comic routine and act powerless. Beg, plead and generally reverse the tables, giving him the feeling of power. Play through it as long as you both want and have fun. He should win, not you. Make sure you end with connection in the form of hugs, verbal "I love yous" or similar emotional closeness and your unconditional acceptance of him.
If you are interested, I got all these ideas from a fabulous book, "Playful Parenting" by Lawrence Cohen.