Arjen Robben tried to single-handedly drag the Dutch to Russia but they fell well short of the improbable seven-goal victory that they needed to make the playoffs.
Sweden finish their Group A campaign as runners-up to France, who edged past Belarus 2-1 this evening, and will battle for a place in the tournament next month.
Robben got the Dutch dreaming of a miracle when he put them 1-0 ahead on 16 minutes, thanks to a helping hand from Manchester United defender Victor Lindelof.
The Sweden centre-back hooked a clearance into his own arm, and the 33-year-old Bayern Munich winger fortuitously scuffed the spot kick past Robin Olsen with a miscued panenka.
On a night when the hosts had to take every half-chance to keep their World Cup hopes alive, Georginio Wijnaldum squandered a golden opportunity when he poked wide from 12 yards out on 25 minutes.
Kenny Tete was guilty of spurning an even better opening on 37 minutes when Ryan Babel whipped an excellent ball across the six-yard box, which the young full-back blazed over the crossbar with a rash finish.
Robben grabbed the Dutch a second before the break with an outstanding first-time effort from Babel's cutback, meeting the ball ferociously with his left foot and bending it inside the far post.
Netherlands ideally needed a three or four goal advantage at half time to really rattle the Swedes, who took the life out of their opponents by surviving the opening 20 minutes of the second period without scare.
The visitors showed no attacking ambition of their own and only had one clear-cut opening throughout the entire contest, when in-form striker Marcus Berg headed straight at Jasper Cillessen from an early set piece.
However, their 8-0 thrashing of Luxembourg at the weekend meant that all the Swedes had to do was avoid a horrific capitulation to finish above the Netherlands on goal difference, and they saw the game out comfortably despite the best efforts of Robben, on what may prove to be his international swansong.
Sweden will find out their opponents in the playoffs next Tuesday and are likely to be unseeded for the draw.