There were so many things that Sega did in the 80's that ensured success in the early 90s. One of these was, when the Genesis came out they still made games for the Master System with "Sega for the 90's: The New Generation" printed on them. These games were sometimes downgraded versions of Genesis games that would still play somewhat the same, only with 2 buttons this time.

Another key element to their success was the right Advertising at the right time. Think about it not many people owned a Computer back then so they were reading magazines and watching TV. For the Genesis, many of the TV ads boasted about 16-bit arcade graphics and blast processing.
Examples:

Magazine:
Bart's Nightmare Magazine Ad (1993)
Sonic the Hedgehog (1991)
Another Sonic the Hedgehog Ad (Early 90s)
Splatter House 3 (1993)
WWF Super WrestleMania (1992)

More Magazine ads here

Television:
Genesis UK Commercial (1992 - 1993)
SEGA Club commercial (1994)
Ecco the Dolphin (1990)
Blast Processing (1993)

TV Spot 'Street Fighter II' Sega Genesis Mortal Kombat (1992)


1993 Sega Genesis Commercial With Sonic The Hedgehog 2 Promo


Mortal Kombat commercial (1992)

Those were the days.

These days, they aren't the days. They're the nights.

I just wish they were more successful. So that way the days could've stayed.