The Dutch, who came third at the 2014 World Cup in Brazil but failed to qualify for this year's European Championship, struggled for long periods against a resolute Sweden side coming to terms with the departure of record goalscorer Zlatan Ibrahimovic.
Having last played at a World Cup finals in 2006, Sweden have recalled former Iceland manager Lars Lagerback as an advisor to new boss Janne Andersson, and the Scandinavians looked compact and organised, if occasionally a little shaky, in defence.
With Sneijder and Kevin Strootman pulling the strings in midfield for the Dutch, Sweden keeper Robin Olsen was called into action early, twice saving smartly from Davy Klaassen before putting in a man-of-the-match performance.
Sweden allowed the Dutch to control possession for much of the first half, seeking to hit them in transition when the opportunity presented itself, and they took the lead just before halftime.
The Swedes howled in vain for a free kick in a good position in the 43rd minute, but the danger was not over for the Dutch as Kevin Strootman then inexplicably gave the ball away to Berg on the edge of the box, and he chipped it over Jeroen Zoet to open the scoring.
The Dutch struggled in front of goal throughout but when Olsen could only parry Daryl Janmaat's shot from distance, Sneijder made no mistake, hammering home the rebound.
Netherlands substitute Bas Dost had the ball in the net again in the final minute but the goal was ruled out after the referee spotted him pushing Victor Nilsson Lindelof in the penalty area, and the game finished in a draw.
Elsewhere in Group A France could only manage a goalless draw away to Belarus, while Bulgaria got the better of Luxembourg 4-3 in a thriller.
(Reporting by Philip O'Connor; Editing by Toby Davis)
