Ronald Koeman named a strong side, including Wayne Rooney leading the line one day on from his surprise England retirement, but the Croatian side hauled themselves back into the tie through Josip Radosevic.
That strike came from almost 35 yards but paled next to Sigurdsson's ambition.
Thirteen seconds into the second half, the Iceland international swung ferociously at a loose ball 50 yards from goal on the right.
Sigurdsson lost his footing but struck the ball sweetly and it dipped beyond helpless Hajduk goalkeeper Dante Stipica.
Everton's focus wavered slightly from that point, with the tie more or less in the bag, and Jordan Pickford – another big-money arrival quickly proving his worth – kept out a 65th-minute penalty from Said Ahmed Said.
Sigurdsson had a nervous moment in the eighth minute when Hajduk midfielder Savvas Gentsoglou went down with him in close attendance inside the penalty area. Referee Sergei Karasev was unmoved.
The hosts made much of the early running before Ante Erceg dragged wide on the end of Said's lofted pass in the 18th minute.
Rooney slipped a lovely pass into Cuco Martina's path on the left but was unable to match that piece of play with an accurate headed finish after continuing his run to connect with the right-back's cross.
Everton were warming to their task and Sigurdsson's eyes lit up when Hajduk failed to clear Dominic Calvert-Lewin's 25th-minute cross from the left, although he sliced a volley wide from the edge of the box.
Pickford saved superbly seven minutes before half-time, tipping Erceg's angled effort to safety after the close-range shot took a deflection off Michael Keane.
But Pickford was left grasping at fresh air when Radosevic's rasping long-ranger clipped Leighton Baines and flew into the net.
The vocal home support inside Stadion Poljud sensed an unlikely comeback but those hopes were extinguished by Sigurdsson's stunning moment of inspiration upon the resumption.
Hajduk's heads took a little while to clear and Rooney might have punished them further when he glanced wide from Ademola Lookman's chipped cross.
Hysen Memolla sprayed an attempt off target as the hosts gamely pushed Everton back, but they were leaving space on the break and Sigurdsson almost persuaded captain Zoran Nizic to put through his own goal with Calvert-Lewin lurking.
The passage of play that led to Ashley Williams diving through the back of Erceg in the box visibly irritated Pickford but the youngster impressively channelled his frustrations, plunging right to deny Said from 12 yards.
Despite those blemishes, Koeman could reflect on an accomplished away European performance that Calvert-Lewin might have added some gloss to when he sprung the Hajduk backline and Stipica stood firm.
Erceg almost capitalised on a mix-up between Pickford and Williams before Rooney checked in to fire at Stipica, by which point Everton's place in Friday's draw was firmly secured.
Key Opta stats:
- Everton have won nine their last 11 European games (inc. qualifiers), losing the other two matches.
- For the second consecutive campaign under Ronald Koeman, Everton have not lost any of their opening six games of the season in all competitions (W4 D2).
- Everton have now progressed from each of their last two Europa League qualifying phases (also in 2009-2010 under David Moyes).
- The Toffees registered more shots on target in the opening 14 seconds of the second half than they did in the entire first half (0).
- Josip Radosevic's opener ended an Everton run of 313 minutes without conceding in European competition under Ronald Koeman.
- After conceding all four penalties he faced at Sunderland (all in the Premier League), Jordan Pickford saved the first penalty he faced as an Everton player.