Holland international Ten Doeschate scored his runs from just 44 deliveries, hitting five fours and five sixes as Essex posted a total of 187 for six after being asked to bat first by David Hussey.
Requiring more than nine an over, the home side began brightly, putting 42 runs on the board in 18 balls before being undone by David Masters.
Hussey, with 61, and Riki Wessels, who made 22, added 63 in eight overs together, but the run rate spiralled out of control and Nottinghamshire's defeat was hastened by Tait, who ended with figures of four for 29 as the innings folded for just 140 in 17.2 overs.
Put in to bat, in front of a crowd of 12,106, Essex were floundering on 46 for three when Ten Doeschate arrived at the crease.
He had some good fortune early on, as a confident shout for lbw was turned down when he had made only six.
Replays seemed to suggest that bowler Ian Butler was unfortunate, with the ball looking to be going on to clip the stumps.
Ravi Bopara made 38 from 25 balls, and shared in a stand of 72 with Ten Doeschate to give the innings needed impetus.
Nottinghamshire's reply got off to a flying start as Lumb and Hales posted 42 in the first three overs before Masters' opening burst completely altered the momentum.
An athletic piece of work by Bopara on the boundary accounted for Wessels and then Tait's final over, the 17th of the innings, effectively wrapped things up as he had Hussey and Butler caught in the deep, before yorking Graeme White to claim a hat-trick.
For the third year in a row Nottinghamshire slumped to quarter-final defeat, while the Eagles, who finished third in the southern qualifying group, now advance to face Northamptonshire Steelbacks in the semi-final at Edgbaston a week on Saturday.
- Sports & Recreation
- David Hussey
- Michael Lumb
- David Masters
- Riki Wessels
- Alex Hales