After a two-and-a-half-hour delay, Adam Milne scored 51 on his Kent debut and put on 79 with Adam Rouse (35).
Rouse was last man to go when he hooked Steven Mullaney (2-44) to Samit Patel as the visitors were all out for 265.
Notts, needing 75 to win, were about to begin their reply as rain returned and the game was called off at 20:30 BST.
Kent, who were at one stage 167-7 in their second innings on day three still needing 24 to avoid an innings defeat, dug in and took eight points from the game with a little help from the weather.
The hosts claimed 12 points and now lead second-placed Worcestershire by 26, with Kent a further 13 points back in third.
Nottinghamshire head coach Peter Moores told BBC Radio Nottingham:
"You can't control the weather, so it's frustrating when you get that close and everybody has put the effort in. Anybody that has been in the game for a while will accept that this was just part of it.
"We got in this position because we bowled them out for under 200 and we got a 200 lead, so the only reason it got twitchy was because we dominated the game."
On the pink ball: "I think there are some issues with seeing the ball at different times, but you can see why the experiment is there - to try and find out whether we can put cricket on at different times of the day when people might find it easier to watch."
Kent head coach Matt Walker:
"I think there has been a fairly mixed reaction to the pink ball. Individually, everyone has had their own little battles with it, I think it's just different, that's the biggest issue.
"It's not done excessive amounts in the air, but it is just a slightly different feel in the hands for the bowlers and sometimes it reacted differently off the surface for the bowlers.
"I think in terms of the batting the biggest issue for the lads is that they weren't able to pick up the seam and weren't quite seeing the shape of the delivery, so they were just seeing a pink blur coming down.
"If they are going to stick with it then I hope they have more than one game a year, so people can used to it, rather than just a token one-off."
Division Two leaders Nottinghamshire had to settle for a draw after resilience from the Kent tail and bad weather denied them victory.