The Irish tired after a superb opening 20 minutes but held firm in a tight Group C encounter at Stockholm's enclosed Friends Arena to claim a valuable point.
Trapattoni's position has been under scrutiny since last summer's poor showing at Euro 2012 but the Irish have still not lost a qualifier away from home under his stewardship.
Sweden had the better chances with David Forde, making his first competitive start, twice saving well in the first half and again in the dying moments to preserve the stalemate.
Ireland dealt well with the threat of Sweden's star striker Zlatan Ibrahimovic and, although Austria overtook them with a convincing won over the Faroe Islands, they will be confident ahead of next Tuesday's game against them in Dublin.
Temperatures in the Swedish capital had plummeted to minus seven degrees by kick-off but it did not matter as the retractable roof was closed. The pitch, covered by an ice rink for a bandy match last weekend, also held up well despite initial fears.
Trapattoni was forced to change his plans before kick-off as Stoke midfielder Glenn Whelan was ruled out with an ankle injury.
Wigan's James McCarthy, who had been controversially omitted in favour of Paul Green in Trapattoni's initial selection, came in to partner the Leeds man in the centre.
Another change to the expected line-up saw Jon Walters come in for Hull winger Robbie Brady.
Brady's place had not been fully confirmed by Trapattoni, with the Italian suggesting he wanted to make sure the 21-year-old was mentally ready, but it seemed a harsh call.
Yet those changes to the team that trained together throughout the week did not disrupt the rhythm as Ireland made a positive start.
Captain Robbie Keane, playing behind strike partner Shane Long, got into a good position but Walters was unable direct a header goalwards.
James McClean then appeared on the right and troubled Sweden goalkeeper Andreas Isaksson with a deep cross.
McClean sent the resulting corner invitingly across goal but Keane completely failed to connect with his shot and the hosts cleared.
Long set off on a mazy run after being fed by McCarthy and found room to shoot outside the area after Andreas Granqvist slipped - but he fired well over.
McCarthy drove Ireland's next attack himself with a powerful run but overhit his pass.
Sweden won a couple of corners in quick succession and got behind the Irish defence for the first time when Kim Kallstrom crossed for Sebastian Larsson - but the Sunderland man volleyed harmlessly across goal.
The Swedes gave warning of their quality, however, when dangerman Zlatan Ibrahimovic headed on and Tobias Hysen connected with a well-directed volley.
Forde showed no sign of nerves with a superb one-handed save to his right.
Forde was in action again soon after when a Kallstrom free-kick from 40 yards out bounced awkwardly in the box but the Millwall keeper palmed wide.
Ibrahimovic, scorer of four goals against England last November, came to life when he found room to shoot from a tight angle outside the box but Seamus Coleman got a foot in the way.
Sweden ended the first half on top by forcing Marc Wilson to head unconvincingly behind his own goal.
Ireland maintained their composure after the restart and it was just after the hour before Sweden seriously threatened again when Behrang Safari ran at the heart of the defence.
Ciaran Clark attempted to cut out his pass but succeeded only in finding Safari again and the Swede's square ball teed up a good chance for Larsson - but again the Sunderland player was wayward, this time slicing wide.
The Swedes won a corner moments later but Forde claimed confidently.
Keane got behind the Sweden line as Ireland looked to snatch a goal but the Los Angeles Galaxy forward overhit a cross and McClean dragged a shot well across goal.
A goalless draw began to look like a good result for the Irish but they were not home yet with Ibrahimovic again threatening but fluffing a header and Alexander Kacaniklic miscuing a follow-up attempt.
Rasmus Elm almost snatched victory in the last minute but Forde produced another fine save to keep out his deflected shot.