Marcus Berg, leading the attack along with John Guidetti for the first time since Zlatan Ibrahimovic's international retirement, put Sweden ahead with a well-taken goal just two minutes before half-time. The Panathanaikos striker capitalised on Kevin Strootman's defensive mistake to chip over Jeroen Zoet into the top corner from outside the box for his first goal in Swedish colours since October 2015.
Robin Olsen, Sweden's new number 1 after Andreas Isaksson's retirement, justified his selection with a string of fine saves to deny the Dutch in a man-of-the match performance. The 26-year-old, of Danish descent who plays for FC Köpenhamn, was though unable to prevent the Dutch equalising in the 67th minute.
Vincent Jannsen's powerful shot was parried by Olsen into the path of Wesley Sneijder to smash home the rebound.
The visitors, who failed to qualify for Euro 2016 in France, had a number of chances to win it and had the ball in the net in injury time but the referee ruled it out for a push by Bas Dost on Victor Nilsson Lindelof.
Holland manager Danny Blind was unhappy with the disallowed goal, he told Dutch outlet NOS:""The referee was in the way. I thought he officiated very well, but this is an incredible mistake by him. We really lost two points here."
Sweden coach Janne Andersson, in his first match in charge of the national team, was pleased with the point and hard-working performance.
"I expected attitude and fight and we saw that from the first minute until the last," he told media afterwards.
He was also positive about his new look midfield of Alexander Fransson, Oscar Hiljemark and Marcus Rohdén.
"There is a lot to work on. To sum up, I place great demands on our own game and keeping more possession of the ball. But it's a good start to the qualifying campaign.
Elsewhere in Group A, favourites France could only manage a goalless draw away to Belarus while Bulgaria edged home 4-3 to Luxembourg.