Is Christianity Exclusivist?  

Interestingly, every religion/philosophy is exclusivistic.   In fact, the Post Modernists and Relativists exclude those who believe God's revelation.   They exclude all who are not Relativists.

Jesus, the Son, and the Father have made very exclusive claims as the following Scriptures show.   If God is the almighty, all knowing, omniscient and only triune God, then those claims make perfect sense.   These claims are simply a statement of fact.

John 3:18  He that believeth on him [Jesus]  is not condemned: but he that believeth not is condemned already, because he hath not believed in the name of the only begotten Son of God.  
John 3:19   And this is the condemnation, that light is come into the world, and men loved darkness rather than light, because their deeds were evil.  
 
John 3:36  He that believeth on the Son hath everlasting life: and he that believeth not the Son shall not see life; but the wrath of God abideth on him.

John 14:6   Jesus saith to him, I am the way, the truth, and the life: no man cometh to the Father, but by me.  
 
Act 4:12  Neither is there salvation in any other: for there is none other name under heaven given among men, whereby we must be saved.