Spiritual experiences can play a very important role in shaping an individual's worldview and religious convictions. Just a single profound mystical experience can dramatically change a person's entire view on life and be the turning point that leads them to embrace an entirely new religious path. Without going too far, the best example in my opinion can be found in Christianity itself, in the conversion of the Apostle Paul, who went from persecuting the Church to becoming the greatest Apostle of all time, all because he had a dramatic "Road to Damascus" encounter (Acts 9).
But spiritual experiences are not exclusive to Christianity. People have reported profound spiritual experiences in all sorts of religions. And people who have had these experiences usually interpret them as evidence and confirmation of the truthfulness of the religion they follow. Of course, whether this interpretative leap is justified or not is a whole different question, but the fact remains that many anchor their religious beliefs in profound experiences they've had throughout their spiritual walks.
And this brings me to the question in the title: How should Christians approach the evangelization of a person who has deep convictions about a different religion because of profound spiritual experiences they've had while following said religion? How to argue against a strong conviction that comes from a profound spiritual experience someone has had in another religion?
______________________
Appendix: examples of spiritual experiences in other religions
1) Hinduism, Buddhism and Eastern Spirituality in general
2)
New Age spirituality
3) Witchcraft & the Occult
- Magic: spell casting, etc.
4) Mormonism