It was only the second time both openers in any World Cup had fallen for first-ball noughts, with Sri Lanka's Lahiru Thirimanne and Tillakaratne Dilshan collecting golden ducks against Afghanistan four years ago.
But they had fallen to two different bowlers Dawlat Zadran and Shapoor Zadran.
After West Indies captain Jason Holder won the toss and put in New Zealand, left-arm quick Cottrell was on target first ball with a delivery that thudded into the pads of Guptill.
Umpire Ruchira Palliyaguruge initially said "not out" but the West Indies successfully reviewed the decision, with replays showing the ball had pitched in line and would have hit the leg stump.
Guptill, who made a World Cup record 237 not out when New Zealand beat the West Indies in a 2015 quarter-final in Wellington, had to go and the Black Caps were none for one.
Misfielding by Carlos Brathwaite allowed New Zealand captain and new batsman Kane Williamson, fresh from his match-winning century against South Africa, to get off the mark with an all-run four.
Williamson then cover-drove Cottrell for three off the fourth ball of the over to leave Colin Munro, his side's other opener on strike.
He too fell first ball, beaten completely by another yorker that smashed into his stumps.
For the second time in the over, Cottrell, still a serving member of the Jamaica Defence Forces, greeted a wicket with his trademark military salute.
Ross Taylor hit a three off the last ball of an extraordinary first over to leave New Zealand 10-2.
( With inputs from IANS )
