I assume you're talking about an 8-speed Shimano rear derailer, and 9-speed Shimano shifters and cassette. This will work. All Shimano rear derailers (except pre-9-speed Dura Ace) have the same amount of travel per amount of cable-pull. So if your rear derailer is 8-speed Dura-Ace, the answer is no.
Your shifting may not be as crisp as with a 9-speed-era rear derailer, because with increases in number of gears, derailers increased their precision as well. So an 8-speed rear derailer may not shift as "snappy" as a 9-speed.

You will also of course need a 9-speed chain, which is slightly narrower than 8-speed.