City midfielder Filippa Angeldahl started for the Swedes and produced an accomplished display before being replaced 20 minutes from time.
In front of 21,342 fans at Bramall Lane, Sheffield, it proved a tough first half for the Dutch who lost captain and goalkeeper Sari van Veenendaal early on through injury.
The 2019 world goalkeeper of the year had to be replaced after being injured coming out for a cross and was replaced by rookie keeper Daphne van Domselaar.
And it was the Swedes who took a deserved lead on 35 minutes with former City striker Kosovare Asllani the architect.
Asllani cut inside from the right flank, nutmegging Dutch defender Aniek Nouwen, before then playing the ball across the box to the unmarked Jonna Andersson.
The Swedish full-back took one touch before slotting the ball past the keeper, with a defender failing to clear on the line.
However, the Dutch responded in kind after the break, deservedly drawing level seven minutes after the restart.
Arsenal star Vivianne Miedema was the catalyst after throwing a dummy and spin past a Swede on the left wing. She then played a quick one-two and then had a pass cut out by a defender.
However, the ball landed to Jill Roord, who drilled a superb low shot into the bottom corner from 16 yards.
Both sides had opportunities to win what was an incident-packed clash but, ultimately, they settled for a share of the spoils.
With Portugal and Switzerland having drawn 2-2 earlier on, the result left all four teams in Group C locked on a point apiece after their opening matches.
Action in the group resumes on Wednesday when Sweden tackle Switzerland at 5pm (UK time) with the Netherlands then facing Portugal at 8pm.