Prolific Arsenal striker Miedema glanced home Sherida Spitse's free-kick to break the deadlock with 20 minutes remaining in the sweltering heat of Valenciennes.
Defender Van Der Gragt secured a deserved Dutch success 10 minutes later, nodding home another Spitse set-piece.
Italy were second best at Stade du Hainaut and twice saved by the woodwork while the game remained goalless.
In the build-up to the match, Holland's request to push back the afternoon kick-off because of the heatwave in France was turned down by tournament organisers.
The searing temperatures resulted in scheduled drink breaks and contributed to a drab, uneventful opening 45 minutes.
Italy's Valentina Bergamaschi directed an early volley straight at Dutch goalkeeper Sari Van Veenendaal following Barbara Bonansea's flick-on, and later scuffed wide when well placed.
At the other end, Miedema's effort from distance and Spitse's tame free-kick were both comfortably claimed by Laura Giuliani.
Italy, playing their first World Cup quarter-final since 1991, were the lowest-ranked team left in the tournament and considered by many as this year's surprise package.
They began to be pressed back in their own half in the second period as the Euro 2017 winners took control of the tie.
Dutch forward Lieke Martens, who scored the last-minute penalty to beat Japan in the last 16 and shrugged off a toe injury to start, twice had sights of goal but lacked the power to beat Giuliani.
Arsenal's Danielle Van De Donk was unfortunate not to open the scoring in the 58th minute with a curling effort from the edge of the box which bounced back off the crossbar following a partially-cleared corner.
Italy continued to ride their luck and were once again reliant on the frame of the goal when midfielder Spitse's powerful free-kick from distance glanced the base of the left post.
However, they could not hold out for much longer and Sarina Wiegman's side duly went ahead with 70 minutes on the clock.
Spitse delivered an inviting, in-swinging set-piece from wide on the left, allowing Holland's record scorer Miedema to flick home her 61st goal for her country and third of the competition.
Italy suffered a further blow 10 minutes later when Barcelona player Van Der Gragt climbed high at the back post to power home Spitse's floated free-kick from the right.
Substitute Daniela Sabatino had a chance to set up a tense finish but her close-range effort was repelled by Van Veenendaal as Italy's hopes of a comeback ended.
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