I had this problem too. Here's what worked for me:
(1) Don't wash your face immediately before waxing. I prefer to wax shortly after I wake up on a weekend;
(2) Spread the wax on thin and use those strips to take it off--even if the wax says you don't need it;
(3) DON'T re-wax an area if you miss something. Putting the strip on there again is okay if you didn't get all the wax off;
(4) Don't wash immediately after either--maybe wipe your upper lip area with a cold wet towel if you've got little blood droplets or something;
(5) Smear neosporin cream on there and leave it on for a while--I walk around the house with white stuff on my face for about an hour;
(6) DON'T DON'T DON'T apply makeup for a few hours. I think this is what gave me those awful zits--and wow, those hurt.
Hope this is helpful to you. If you're still having problems, try getting the wax professionally done. It may be the way you're doing it. I learned, the hard way, that I can't do my own bikini waxes for the same reason. Grrrr.
Having to pay for a wax (whether bikini or upper lip) is definitely better than walking around with those awful zits and ingrown hairs. Good luck!