Saturn came out a year earlier than the PS1, but the Saturn started at $399. PS1 came out at $299. Sega was already losing money at the $399 price due to adding an additional CPU to try and combat the graphics of the PS1. Sega was not prepared for the 3D era and as a result, the Saturn pretty much flopped. It was possibly the hardest system to program for in history (it's actually still one of the hardest systems to emulate on a pc). Sega shot themselves in the foot beforehand. They reacted to Sony and in a bad way and it cost them. They were out of the race before FF7 was announced and before the N64 was released. The N64 was their final nail. In fact, if the PS1 never existed, the N64 would have crushed the Saturn because the Saturn would have been only marginally more powerful than the SegaCD.
Of course, there's a chance that if the PS1 didn't exist, the Saturn would not have been rushed to market and may have had specs closer to the N64 as they would have had at least another year of development before releasing it.
Revisionist history is hard to do.