Jesus responded to the Canaanite woman in the way that He did "Fulfilling law!"
Galatians 4:4 says "...God sent forth his Son, made of a woman, made UNDER THE LAW..."
Do not be deceived! The first four books of the New Testament were written "under the law!" Just because "man" has put Matthew, Mark, Luke and John under the "New Testament" does not mean that they were under the 'New Covenant!" How could they be when Jesus came to "fulfill law" and those books were accounts from disciples remembering their accounts while walking with Jesus? If Jesus was still ALIVE and had not yet "shed His blood" nor sent promised Holy Spirit, the law had not been fulfilled!
Until Jesus DIED and sent promised Holy Spirit (promised in John 14:26 and sent in Acts 2:2), the New Covenant (Acts) was not effective and previous New Testament writings were "under the law!"
Under New Covenant we (everyone, not only Jews) are NOW able to inherit eternal life! It is BY FAITH that we (ALL) NOW have the right to "eternal life" which is not only for Jews!