Jesus responds to the woman in what might seem like a severe way. But their interaction shows that Jesus wanted to see her faith (compare Matthew 15:28).
Jesus did not use the word first to exclude her as a Gentile (non-Jew) from God’s grace (kindness and favor). Instead, it showed that his mission was directed first to the Jews (Acts 13:46; Romans 1:16).
The woman replied with humility and faith. She acknowledged the priority of the Jewish people but argued that there was more than enough food for the dogs as well.
Jewish writings often referred to Gentiles as dogs. Dogs were seen as dirty scavengers rather than family pets (see also study note on Revelation 22:15).
The woman’s persistence and acknowledgment of Jesus as Lord were rewarded.