Millar lobbed keeper Nikita Haikin with seven minutes remaining at Windsor Park to secure a priceless 1-0 win for the Northern Irish champions.
The Norwegians, who thumped Jose Mourinho’s Roma 6-1 in last season’s Europa Conference League, will have to make home advantage count next Wednesday if they are to progress.
Linfield had to defend tigerishly after a tight first 45 minutes, with Ben Hall and Stephen Fallon making crucial second-half interventions to deny Joel Mugisha and Amahl Pellegrino respectively.
But Millar took full advantage with his 83rd-minute strike to pile the pressure on Bodo/Glimt when the sides reconvene at the Aspmyra Stadium for the right to face either Zalgiris or Malmo in the next round.
Pieros Sotiriou’s first-half double left Shamrock Rovers with a mountain to climb as Ludogorets took a firm grip on their tie.
Rovers found themselves under pressure from the off and fell behind with 25 minutes gone when Sotiriou headed the home side in front, and Rovers’ woes deepened within 10 minutes when the striker added to his tally with a right-foot shot.
Dylan Watts fired just over as the visitors responded after the break and substitute Graham Burke came close to reducing the deficit with five minutes remaining.
However, Ludogorets substitute Igor Thiago ensured the Bulgarians will take a 3-0 lead to Dublin next week when he struck deep into stoppage time.