Worcestershire 338 for 7 (Haynes 153, D'Oliveira 123) beat Essex 156 (Wheater 77, Cook 42, D'Oliveira 3-8) by 182 runs
Haynes was flawless in his maiden professional ton of 153 while his opening partner's 123 made him the third generation of D'Oliveira to reached three figures for Worcestershire in the format.
Their stand of 243, a best for the first wicket for the county, laid the foundations of a mammoth total of 338 for 7.
Essex collapsed to 17 for four in reply, and despite Adam Wheater's 77, were bowled out for 156 and lost by 182 runs - their highest ever defeat.
The Rapids maintained their 100 per cent record in the tournament while the Eagles lost for the first time.
Having been put in by Essex, D'Oliveira and Haynes strutted against a depleted home bowling line-up - with Sam Cook at the Hundred and Jamie Porter injuring his side during the victory over Middlesex.
D'Oliveira won the race to a half-century, smashing his 54th ball into the Tom Pearce Stand at wide long-on, before Haynes followed in 57 deliveries - for Haynes it was his third fifty in as many innings in the competition.
D'Oliveira also reached his hundred first, off 95 balls, but Haynes was quicker, in 91 balls, as both passed the milestone for the first time in professional white-ball cricket.
The incredible stand ended on 243, a best for Worcestershire of any stand against Essex, when D'Oliveira picked out wide long-on.
That started a wicket tumble as Rapids lost seven wickets for 68 runs in 69 balls.
Joe Leach was lbw to Aron Nijjar, Tom Fell was caught on the square leg boundary, Jake Libby was brilliantly run out by Simon Harmer and Ed Barnard was caught behind attempting a reverse sweep.
Haynes continued to accelerate, bunting over the long-off boundary and a sweet strike straight down the ground for two sixes - the latter off Shane Snater, who returned Essex's third most expensive List A bowling figures with his one for 83.
The only chance he offered was a diving effort to Jack Plom at backward point on 151 before he was finally stumped off Harmer for 153 off 128 balls.
Gareth Roderick scooped to fine leg but Joshua Dell's 27 off 24 helped Rapids back up to a total befitting the opening stand.
Essex's top-order collapse was even more emphatic as they slumped to 17 for four in the first six overs of the reply.
Will Buttleman was yorked fourth ball by Charlie Morris before Joe Leach has Tom Westley bowled, Michael Pepper feathering behind and Ryan ten Doeschate leg before to an in-ducker.
Wheater was dropped on 16 and 23 both by Dell at cover during a 70-run stand with Alastair Cook before Ed Barnard took two wickets in two balls - Cook clothing to mid off for 42 and Harmer caught behind - to keep the win predictor firmly pointing towards the visitors.
Wheater accelerated after Cook's dismissal but eventually holed out to long-on, Nijjar holed out to long-on, Snater chipped to midwicket and Plom was stumped - the last three wickets falling to that man D'Oliveira (three for eight).