Marc Skinner's United dominated for long periods in Norway but were hit by Ingrid Stenevik's close-range header in the 76th minute from a Signe Gaupset free-kick.
The English club will need to overturn a one-goal deficit in the return match in a week's time to reach the inaugural 18-team league phase.
United have never played in the tournament proper before.
Real Madrid, who lost to eventual champions Arsenal in last season's quarter-finals, will defend a 2-1 lead next week against Eintracht thanks to goals from Sweden midfielder Filippa Angeldal and Denmark's Signe Bruun.
Former semi-finalists Paris FC were held to a goalless home draw by Austria Vienna, while Atletico Madrid drew 1-1 with Hacken in Sweden.