Rather than just say money:

1). Microsoft hasn't released 2 major additions to the 360 with games which aren't cross compatible (Genesis Mega CD and 32x).
2). Microsoft hasn't released a major console on the day of its full E3 reveal (Saturn).
3). Microsoft hasn't killed off it's old console while waiting to build up to its new release (Saturn --> Dreamcast).
4). Microsoft isn't trying to incorporate a design which is unheard of at it's time (Saturn's Dual Core).
5). Microsoft didn't have two separate teams designing two totally different consoles on different consoles (Sega: USA 32x and Japan Saturn IIRC).
6). Microsoft hasn't dismantled it's American game studios (well not quite Sega levels anyway). (Sega of Japan had enough of Sega America and basically closed it down, so Sega's overall game output was roughly reduced by half).
7). Microsoft doesn't have a handheld.
8). Sega didn't have Microsofts money :P

For the original xbox, it is a mix of things but above all it was money.

Money

• They entered the market with a powerful console that was priced reasonably (unlike 3DO or Neogeo).
• The marketing for the system was huge (unlike the apple Pippin for example).
• They attracted western 3rd party support with attractive money-hatting proposals (unlike Nintendo's cube).
• They took measures to pierce into the japanese market no matter how hard that was. (Mystwalker and others)

360

• They were able to do loss-leading and sustain the console until profitability (unlike dreamcast).

Other

• It was familiar to PC devs and easy to port games between it and the PC: directX technology (unlike say the Saturn).
• A good online infrastructure early on. (This also involves money, but is strategy)
• They secured key exclusives like Halo and DoA (this also involves money, but is strategy)
• Tighter management. Better business decision-making.

• They had moderate success in cutting into Nintendo's casual market.
• Good branding with the 360.
• Controller and console design aesthetics/ergonomics issues with the original xbox were fixed. Best trad controller in the industry.
• They re-launched the console with a redesign and a major peripheral upgrade (kinect).

Money.

They have a huuuuge amount of money to spend on such things

Money was a big factor. Though I say it has to do more with Peter Moore (Or if there was someone behind him controlling things) than money. Microsoft would just be surviving if it was just money. Their popularity came from Moore's aggressiveness. Sega probably would still be making hardware if Sega of Japan wasn't filled with old guys who don't understand games (Hint hint Nintendo and old Sony). Hell most of the Genesis' success was because of Sega of America (Which Moore headed up.) We more or less have to see what Microsoft does now, if rumours are true it sounds like they're following Moore's leg work instead of their own.

You are a harsh mistress, but a fair one.

Money is obvious. Not so much how much Microsoft has but how much Sega didn't to recoup the mistakes they made with the Saturn and before. The dreamcast was great and was actually selling pretty well at the time. If they could have stayed afloat financially and pushed out more games they could have competed with PS2 as the more affordable option with better internets, after reducing their internet fees of course.

His whole head is a bald spot. I guess that's a lot of luck.

Microsoft was sitting on an enormous amount of money that could absorb the massive costs of starting up. Sega was a gaming company only