With southwest side Somerset defending a relatively modest total of 145, paceman Henry led the attack with 4-24 from 3.3 overs, while leg-spinner Sodhi took 3-22 from his maximum four overs as Essex were dismissed for 131 with nine balls to spare.
Adam Rossington and Dan Lawrence made a rapid start to Essex’s pursuit of a target of 146 only for Henry to dismiss both batsmen before bowling Michael Pepper with an excellent delivery.
Essex were struggling at 55-5 and were 80-6 when Sodhi bowled Paul Walter.
Daniel Sams, however, threatened to win the game off his own bat for Essex with a rapid 45 off 26 balls.
But just when the east of London club were eyeing a remarkable victory, Sams slashed Henry towards third man for what looked like another boundary only to be brilliantly caught one-handed by a diving Tom Kohler-Cadmore as Somerset won their first Blast title in 18 years.
“[It feels] pretty damn good, I’ve been to finals day many, many times, and this is a special feeling to get over the line tonight,” Somerset captain Lewis Gregory told Sky Sports.
Essex skipper Simon Harmer said: “At the end of the day we were outplayed. I think they (Somerset) bowled exceptionally. Unfortunately, sometimes there’s a loser and that’s us tonight.”
Earlier on finals day in Birmingham, Somerset beat Surrey by 24 runs in their semi-final while Essex defeated defending champions Hampshire by five wickets under the Duckworth-Lewis-Stern method for rain-affected matches.
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