No, it didn't finish three years into its five year mission. There's no indication that one season = one year on the show. It's just a catchy phrase, with no actual significance, just like "a three hour tour" for Gilligan's Island.
The Korean war lasted about three years and a month. MASH, the TV series about the Korean war, lasted 11 seasons (years), 255 episodes and the 2-1/2 hour finale. I think if Roddenberry had his druthers, the five year mission would have stretched out over ten seasons, or more.
On this site, someone actually tries to put structure and logic to the stardates on the different shows. He extrapolates about 1.75 earth years covered between the earliest and latest referenced star dates over the three seasons. He even has a "star date" calculator for any date you want to put in, in TNG stardate format.