TL;DR - Jehovah's Witnesses don't believe they are the only people who will be saved. Jesus' ransom sacrifice was designed to retroactively cancel out the sin which has been inherited by everyone from Adam. (1 Corinthians 15:20-22; Hebrews 2:9) Sincere ones who have died without an opportunity to exercise faith in Christ will receive an opportunity when they are resurrected during Christ's Millennial Reign. (John 5:28-29; Acts 24:15)
First off, you may want to read the article: What Is Salvation?
Just as we are able to put faith in Jesus' sacrifice even though we didn't personally see it, Jehovah's people had been putting faith in the Messiah long before Jesus brought about the prophecy's fulfillment. (John 8:56) When Adam and Eve first sinned, God immediately spoke of a Messianic prophecy by which humankind would eventually be restored to a sinless state.
And I will put enmity between you and the woman and between your offspring and her offspring. He will crush your head, and you will strike him in the heel.” —Genesis 3:15
Throughout history, faithful men and women have been putting their trust in this promise as God continued to expound on how it would be fulfilled by the Messiah. Hebrews chapter 11 lists many faithful ones from the Old Testament who found God's favor and received salvation.
13 In faith all of these died, although they did not receive the fulfillment of the promises; but they saw them from a distance and welcomed them and publicly declared that they were strangers and temporary residents in the land. 14 For those who speak in such a way make it evident that they are earnestly seeking a place of their own. 15 And yet, if they had kept remembering the place from which they had departed, they would have had opportunity to return. 16 But now they are reaching out for a better place, that is, one belonging to heaven. Therefore, God is not ashamed of them, to be called on as their God, for he has prepared a city for them. —Hebrews 11:13-16
However, all of these faithful ones from before Jesus' day have the earthly hope, just as most Jehovah's Witnesses have today. The heavenly hope has been reserved for 144,000 which have been chosen by Christ ever since his death. Those with the heavenly hope will assist Christ in ruling the righteous ones on Earth.
39And yet all of these, although they received a favorable witness because of their faith, did not obtain the fulfillment of the promise, 40 because God had foreseen something better for us, so that they might not be made perfect apart from us. —Hebrews 11:39-40
This is the only distinction between those who died before Jesus' death and those who died after: the ones who died after are able to be members of the 144,000.