
Vergil stabbing Dante with his own sword in DMC3 was a pretty cool story point, for a number of reasons.
Vergil inadvertently creates his own nemesis and the very being that will in the end defeat him.
Prior to the fight, it was clear as a thousand of suns that Vergil was the superior guy between the two. He beats Dante with seemingly no effort, even going as far as mocking him, with words and actions.
Stabbing him with his own sword was an act of haughtiness to humiliate Dante even in death, but it ended up unlocking Dante's potential and releasing his DT, thus beginning his path of gaining power.
Yep. That's another cool narrative point right there. I'm sure a lot of people have caught the irony of it: between the two brothers, Vergil is the one chasing power at all costs, yet he was also the one who allowed Dante to gain it, eventually denying it for himself.
Quite a nice result, considering that that could have been the end for Dante.

Obviously, being a prequel made after DMC1 and 2, we knew Dante was to survive, since he's alive in those two.
But anyway, ever thought about that? This moment is what allowed the other games's events to happen. Or to phrase it better, Vergil is the one who allowed the other games's events to happen. By creating his own nemesis and giving him the weapons to beat him, and to pull off all the feats Dante pulled off in the other titles.