The high harmony is not necessarily sung in falsetto--generally it is sung is what is known as the "head" voice--surprisingly, a least with work, you can hit the same notes in your head voice as in your "falsetto", but you will have a lot more control over expression and timbre in your head voice--

You must have some aptitude, or your bandmates wouldn't have given you the part--the real trick then is to figure out what your range is and then to move the keys around on every tune you do so that you never have to push yourself outside your range--other thing to note is that often, in the high parts, the melody is altered so that the high voice doesn't go too high--

Beyond that, remember that the object is to get as much sound from as little air as possible--also, as someone mentioned above, very important to keep the muscles relaxed--this, more than anything will extend your range and keep you from damaging your voice--

Another thing to note, though your larynx always makes the sounds, depending on where you are in your range, the sounds are controlled(and resonate) in very different places, from deep in your chest all the way up to the front of your hard palate and your teeth--it is important to learn to move smoothly from one area to another, and, arragement wise, recognize that the voice quality changes when you move from one area to another, and to write the part so that it stays in the range that sounds best--

Another thing to remember, the male voice descends with age--The young Frank Sinatra was often described as "adenoidal" but as he got older, that voice had deepened and broadened out to about the best baritone voice ever recorded--if you are older, you'll do better, if, rather than trying to copy vocal arrangements done by teen and twenty year olds, you reconceptualize them to reflect your range--