World champion Kenny held his nerve in a tight finish, as last year's winner Crampton settled for second and Lewis Oliva completed the podium.
"It was nitty gritty on Matt's (Crampton) wheel in third place. I was just happy to hold station knowing Matt was one of the race favourites," said Kenny.
"The national Keirin championships is always tight so going to the front quite early on I was a bit of a sitting duck.
"Lewis (Oliva) went down the inside and pushed me out wide so I had no choice in going to the front."
Elsewhere, Jess Varnish and Laura Trott claimed their second titles of the week in the women's sprint and points race respectively.
Varnish edged out Victoria Williamson while Trott again bested London 2012 team-mate Dani King.
"It was really hard, it was a tough race; it's not like a pursuit, that's predictable, but in the points people use different tactics so you never know what's going to happen," said Trott, the double Olympic champion.
Sam Harrison took the men's scratch race ahead of Olympic team pursuit gold medallists Ed Clancy and Steven Burke.