Kayaba overrides the system to let Kirito and Asuna live. Remember, people don't die IMMEDIATELY upon 'death' in the game. They have at LEAST 10 seconds(and I would suspect its actually quite a bit longer) before they are actually terminated, otherwise the item from episode 3 or 4(the revival item) wouldn't work.

So, when Kayaba was defeated, he overrode the system and revived them as a reward for completing the game.

I've always suspected that the place they go to talk to Kayaba after winning was sort of a 'reflection' place that everyone who dies goes to before actually fading so they can come to terms with their deaths.

Edit: As to how Kirito managed to 'overcome death' at all and how Asuna managed to overcome paralysis... I have theories about those. My wife and I talked about this at length.

The first thing to accept is that every player's mind is directly linked into a computer system. This is fact, that's how the whole thing works. The second piece of knowledge that is required for this is that Kirito is a bit of a computer savant - this is established in a few places, but most notably when he hacks the gm console for Yui. So we know Kirito knows his way around computers, game design, etc. The third thing is that they have established a precedent for 'player will' to have some impact on the game - established by Kirito slightly resisting the paralysis(allowing himself to move small amounts such as pulling the blade out slightly) in an earlier episode. It could be argued that there was no actual 'will' stat, but that the equivalent benefits provided in a typical rpg by that stat were granted by a player's determination as a person. IE. the ability to resist status ailments.

Asuna's Paralysis - In desperation, both over the thought of losing Kirito AND the thought that she'd have to live without him(the suicide prevention Heathcliff promised), her 'will equivalent' was too much and the paralysis couldn't hold her. Since the ability to partially resist paralysis was established, the ability to overcome it is not too far of a stretch.

Kirito's 'Victory' over death - I believe that Kirito's mind subconsciously HACKED the game in desperation over the thought of his death and his failure. Remember, he's an expert with computers and programming and the like, so removing the limitations of 'fingers' from the ability to interface with the game, he was able to break into the servers and reconstitute his code temporarily. Afterall, if his mind was DIRECTLY linked into the server - which is what the NerveGear supposedly does - why wouldn't it be possible for him to do so? And, as to why he might not have done so previously... never before did he have so much at stake. As to why it wasn't permanent, the system has code for automatically detecting glitches and terminating them - as seen by Yui's 'deletion', so Kirito was only able to 'revive' for a few seconds before Cardinal caught him and reversed his 'revival'. The argument might be made that 'it's never happened before' but do we really know that? For example, if someone had committed suicide by jumping off the castle, and then reconstituted would we have ever seen it? They'd have died again before anyone could notice. Or say someone stumbled upon a boss and was killed instantly, if they revived themselves they would've never been seen again still since the system would've caught them before they could act.

So that's my theory :)