We do know that the Bible tells us to do so.
I Tim 2:1 “I exhort, that first of all supplications, prayers, intercessions and giving of thanks be made for all men… for this is good and acceptable in the sight of God, our Savior, Who will have all men to be saved and to come unto the knowledge of the truth.”
We also have the example of the Apostle Paul in that he prayed for the salvation of people.
Rom 10:1 “My hearts desire and prayer to God for Israel is that they might be saved.”
It may be difficult to synchronize how our prayers work in relation to the Word of God and the Spirit of God in this process but until we understand it, we should just pray. A great resource to help people combine prayer and evangelism is: http://prayingforyou.org. They have a resource that is a 7 hour training seminar on this topic. It would be a great tool for a small group or Sunday School to study this topic.