Beginning their quest to defend their 2022 crown, the Lionesses broke the deadlock in League A Group 3 through Alessia Russo but were otherwise unimpressive, and Peter Gerhardsson's earned a merited stalemate through Fridolina Rolfo's second-half effort.
Opting to make just the one change to her first XI from February's 5-1 friendly thrashing of Italy, Sarina Wiegman introduced Lauren James back into the starting lineup at the expense of Chloe Kelly, while the returning Leah Williamson took her place on the bench.
The reigning European champions saw little of the ball in the earliest moments, but Sweden's possession dominance did not translate into any clear-cut openings; a lengthy stoppage for a Russo injury did not help the Scandinavians.
The Arsenal forward was eventually cleared to return to the field, and just four minutes after Rolfo's attempt from a tight angle flashed the wrong side of Mary Earps's far post, Russo broke the deadlock against the run of play.
The ever-dangerous James was unsurprisingly the instigator, working a yard of space inside the Sweden box and sending a delicate ball into the mixer, which Russo met with a close-range diving header to propel England ahead.
Wiegman's side lacked attacking ingenuity otherwise and reverted to holding their lead for the remainder of the first half, although a couple of promising Sweden counter-attacks petered out before the hosts trudged down the tunnel a goal to the good.
The hosts were still bereft of ideas in the final third as the second half began, and while they were not punished for their uninspiring play in the opening period, Sweden finally found the mark with 64 minutes on the clock.
Again, a delivery from the right-hand side of the box did the damage, as Rosa Kafaji's cross was on the money for Rolfo, who headed through the legs of Earps at the back post to draw the visitors level.
The lukewarm Lionesses were almost caught cold again in the 68th minute, as Stina Blackstenius beat Arsenal teammate Lotte Wubben-Moy in a foot race and found herself one-on-one with Earps, but she dragged her shot centimetres wide of the far post.
Sweden kept the hosts at arm's length effectively as the clock ticked down, barring a slight scare in the 88th minute, as Jennifer Falk denied Lauren Hemp before Magdalena Eriksson got in the way of the Manchester City attacker's follow-up.
The Lionesses - who sit third in the group after one match - now have three full days to recuperate before a trip to Dublin to face the Republic of Ireland next Tuesday, while second-placed Sweden host France in Gothenburg on the same evening.