The Croatian side had their tails up at the start of the second half having halved the aggregate deficit two minutes before the break through Josip Radosevic's 30-yard rocket that swerved past Jordan Pickford.
Their home faithful were silenced just seconds after the restart when Sigurdsson, making his full Everton debut, beat goalkeeper Dante Stipica with an audacious shot from almost 50 yards.
His goal took the wind out of Hajduk's sails and their exit was virtually sealed when Jordan Pickford saved Ahmed Said's penalty.
“Personally, it was fantastic to get my first goal for the club — and in a fantastic stadium and fantastic atmosphere as well,” Sigurdsson told evertontv.
“I probably did not expect to score from there. But I managed to get a touch on the ball — it was sitting up quite nicely and I thought to myself, ‘the keeper is probably not on his line'… so I decided to go for it and, thankfully, it beat the keeper and hit the target.
“We are going home happy, which is the main thing. The most important thing was to get the result and be in the hat for the draw.”
Having only played the last half hour or so at the Etihad Stadium on Monday, Sigurdsson probably wasn't expecting to play the entire game but his manager eventually kept him on the field until the end.
His protracted move from Swansea City meant that the 27-year-old missed almost all of pre-season and he is now in catch-up mode as the season gets going.
"It was my first start since pre-season and my first 90 minutes," Sigurdsson continued. "I am a bit tired but I really enjoyed it. It was a good night for me personally and for the team.
“The manager kept saying to me, ‘10 more minutes, 10 more minutes'. He must have said it three or four times. Then I looked at the clock and it said 85 minutes. But at the end he said, ‘you needed the minutes'!
“But the last 10 minutes wasn't too difficult, we were keeping the ball really well and making them chase it. It was pleasing for me to get through it, get 90 minutes under my belt and get the goal as well.
“It was a great night of football, the home and away fans were terrific. These are the kind of nights you want to be involved in, in big stadiums, with great fans.”