One good thing is to use the R-27ER (If I remember correctly that's what he used in his tutorial), or your long stick. Launch it at him, and follow the tutorial. More than likely however, he's going to not get hit by it, which is okay. After firing, go higher by F-Poling. It puts you at a higher position and makes his missiles shorter ranged, while the incoming missile puts him at a disadvantage, because he has to react. So while you're F-Poling, watch the data in the HUD. If it starts to go crazy, that's him jinking. If the data starts to stop going crazy, he's coming for you. Pop him with an R-27ER and that might sweeten the deal. However, you're more than likely going to at least take an AIM-120/7 shot. I always preferred diving, combined with heavy use of chaff. Most of the time it works, but you have to consider keeping him locked up until you kill him.
I think another main problem is that you're not using your missiles right. The Su-27 by default has SARH, or missiles that require you to maintain a lock on the target in order for them to kill. You can also carry IR missiles, but usually the best ones are your R-27Rs (I would say -ER, but now it's just useless) or even R-77s if you add them. The key thing is to keep the Eagle busy not firing, while you are firing, which may involve a waste of missiles, but the good thing is that you won't have to worry about the taxpayers griping about waste :D