The Red Devils replaced Roberto Martinez with Tedesco after exiting the FIFA World Cup at the group stage, while a number of big names also retired from international football.
Tedesco's tenure got off to a flying start at Friends Arena in Stockholm, where Lukaku's header and a close-range finish either side of half-time had the visitors in control.
Sweden brought on 41-year-old striker Zlatan Ibrahimovic, but Lukaku completed his treble in the 82nd minute to round off a comfortable victory.
Dejan Kulusevski almost gave Sweden the lead when his shot trickled towards goal after Thibaut Courtois got a hand to it, only for Wout Faes to clear off the line.
It was the visitors who opened the scoring 10 minutes before the interval when Dodi Lukebakio's right-sided cross was glanced in by the head of Lukaku.
The on-loan Inter Milan striker doubled his tally with a tap-in from a yard out after Lukebakio burst into the box, got away from his man and once again picked out his team-mate.
Sweden struggled to ask any real questions of Courtois, despite the introduction of Ibrahimovic, and Lukaku made certain of the win when given a simple finish by Johan Bakayoko.