There's two ways that I know of, but only one is consistent and reliable: son1cgu1tar's method. While you're going down the slope, uncurl from your spindash shortly before you hit the spring and start holding B again before you hit it, as if you were going to charge another spindash. Keep holding B until Shadow grabs the vine. This should consistently throw you past the spring.
If you're wondering, the other method (mine) is to start mashing pause while Shadow swings around on the vine while wildly moving the control stick in random directions and keep doing so until he lets go of the vine. I only got about a 40-50% success rate with it so once I found out about son1cgu1tar's method I completely quit using mine (but if you want to see it in action one of my runs on Youtube demonstrates it).
P.S. I don't know if either method works on the HD version of the game as I don't have it, but they work on the Gamecube and AFAIK Dreamcast versions.