Ryan ten Doeschate had hit a career-high 119 to lead the unfavored Dutch to 292-6 but opener Andrew Strauss gave his team a platform with 88 from 83 balls, equaling the highest score by an England captain at a World Cup.
Jonathan Trott hit 62 at almost a run a ball but England still cut it fine, with Ravi Bopara hitting the winning runs with just eight balls remaining to take England to 296-4.
Ten Doeschate's 110-ball innings, which included three sixes and nine fours, and a conservative chase on a good wicket, suggest England could struggle in its next Group B match against tournament favorite India on Sunday.
Ten Doeschate, one of only a handful of professionals in the Netherlands' squad, followed up a century that featured a spectacular sixth-wicket partnership of 61 from 32 balls alongside captain Peter Borren by taking two wickets to keep England pegged back until late on.
Barresi stumped Trott with a quick take before Ten Doeschate took out Ian Bell's middle stump to dismiss him for 33 and leave England at 241-4.
England still needed 13 from the last two overs but Bopara eased the pressure by hitting his team's first six off Bernard Loots over long off to reduce the required rate drastically.
That seemed to lift a weight from England, and Bopara struck two more runs before a pair of boundaries clinched victory with the No. 6 on 30 from 20 balls.
There were some positives for England, with captain Strauss and Kevin Pietersen getting the reply off to a flying start, but it needs to cut out the uncharacteristic mistakes that allowed the Netherlands to post its best total against a test nation at a World Cup.
Also the second highest total by any team against England at the World Cup, it surpassed the 230-6 the Netherlands made against the same opponent in 1996.
Pietersen dropped a difficult chance from Cooper - although the batsman fell to Paul Collingwood next ball - and Ten Doeschate reached his century off 98 balls when an attempted run out ricocheted off the stumps and went for four overthrows.
Graeme Swann meanwhile dropped Borren, who made an unbeaten 35 from 24 balls, in the final over.
As bad as they seemed, there were even worse errors on show from England's fielders.
Ten Doeschate should have been out for 47 but a spectacular miscommunication between James Anderson and Pietersen meant neither went for an easy catch and the pair let the ball drop safely between them.
Borren was bowled in the penultimate over - only to be recalled from the rope when it was shown England had only three fielders inside the circle.
Any more mistakes and England would have endured a repeat of 2009's humiliating World Twenty20 defeat at Lord's against the Dutch.
