Chasing 201 for victory, Wayne Parnell (3-33) reduced Middlesex to 11-2 and Denly's inspired spell secured victory.
The highlight was a superb one-handed return catch to dismiss James Franklin.
Earlier, Sam Northeast's 55 helped Kent post 200 all out after Toby Roland-Jones (3-35) removed in-form opener Daniel Bell-Drummond for a duck.
Bell-Drummond had scored centuries in his previous two One-Day Cup outings against Somerset and Sussex, but his stay at the crease on Sunday lasted for only one legitimate delivery.
Roland-Jones' first ball was a wide, but Bell-Drummond edged the second behind to wicketkeeper John Simpson to fall without scoring.
After a shaky start to their reply, Nick Compton (37) and Franklin (33) looked like getting Middlesex close to Kent's total, but Compton's dismissal by James Tredwell sparked another collapse of four wickets for 22 runs.
Joe Denly was an unlikely hero with the ball, taking 3-20 as Kent bowled out Middlesex for just 154 to earn a One-Day Cup win by 46 runs at Canterbury.