The European champions had struggled for goals in their lead-up matches and looked similarly toothless against the world No.53, who rose to the occasion on debut in front of 44,369 fans at Suncorp Stadium.
Only Georgia Stanway's 29th minute penalty was the difference.
Stanway converted on a second attempt after Haiti goalkeeper Kerly Theus saved her first spot-kick but was ruled to have encroached.
Dumornay, 19, gave England headaches throughout while Lionesses skipper and centre-back Millie Bright looked rusty and Mary Earps made two big saves.
Haiti defender Jennifer Limage was forced off after half an hour with seemingly a serious knee injury.
In the 14th minute, Dumornay pounced on the ball after it dropped from an aerial contest and slipped through Roselord Borgella but she dragged her shot wide.
England thought they had a penalty when Chloe Kelly was left with a scratch down her leg from a challenge from Dayana Pierre-Louis.
After a video assistant referee review (VAR), it was determined Alessia Russo had fouled Tabita Joseph in the build-up.
A brain snap from Haiti's Batcheba Louis gave England their chance from the spot.
Inexplicably while jumping, she threw both arms in the air and the ball clipped her hand.
Stanway's penalty was poor, but after the VAR intervention, she made no mistake with her second effort into the bottom corner.
England were wasteful with their chances while Dumornay was a constant thorn in their side.
Five minutes into the second half, Dumornay dispossessed Lucy Bronze, burst forward and ripped a long-range strike barely denied by Earps.
Russo forced back-to-back saves from Theus in the 58th minute while the goalkeeper tipped the striker's header over the bar six minutes later.
Haiti never stopped and Earps denied Roseline Eloissaint with a flung-out leg in the 80th minute to save England all three points in Group D.
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