Hammarby had begun their Group 2 campaign with a 2-0 win over Brage, and tonight's trip to another lower league side began in similar fashion when Kabore put them ahead, only for Orebro to respond through Christopher Redenstrand.
Besara restored the advantage before the break, then he and Kabore looked to have put the Stockholm side out of sight by both completing their braces in the first 20 minutes of the first half.
Yet from 4-1 down, the Superettan side almost pulled off a comeback, as Antonio Yakoub reduced the deficit almost immediately, then Hampus Soderstrom made it a one-goal game with seven minutes to play.
However, Hammarby finally secured the points five minutes from time as Besara completed both his hat-trick and the scoring in the 5-3 win, taking Bajen to the top of the group, two points clear of Oster, for whom Filip Olsson also scored three in a 4-1 victory over Brage.
In Group 3, Sandsviken were on course to cause the upset of the round when they went 2-0 up against GAIS, only for the Allsvenskan side to turn things around and run out 3-2 winners away from home.
Christian Wagner and Alan Carleton had the Superettan side two goals ahead inside 15 minutes, but the match swung back in the Mackerels' favour two minutes into first-half added time, when Filip Beckman reduced the deficit.
That confidence boost before half-time proved vital for the visitors, who equalised through William Milovanovic, before Henry Sletsjoe completed the job with 15 minutes to play.
GAIS are top of the group with four points, ahead of Norrkoping on goal difference after they won 2-0 at home to Landskrona. The top two will battle it out for a place in the quarter-finals at the weekend.
Group 5 witnessed the gulf in class between the Allsvenskan and the fourth tier Division 2 - Norra Gotaland, as Djurgarden hammered IFK Skovde 8-0. Nino Zugelj and Oskar Fallenius scored twice each as the Iron Stoves scored four in each half for their first win in the group.
Djurgarden will need to beat Brommapojkarna to progress, as BP won 4-1 at home to Falkeburg to stay three points clear, albeit with the same goal difference as the Stockholm side.
In Group 8, Sirius and Elfsborg will face off next time out for a quarter-final place as they followed up big wins in the opening round with Sirius beating Helsingborg 4-1 and Elfsborg winning 1-0 at Sundsvall. Better goal difference means the Uppsala side only need a draw to progress.
There will also be a winner-takes-all clash in Group 6, where Malmo sit top, ahead of Halmstad on goal difference as both maintained perfect records so far. Erik Botheim scored twice in MFF's 4-0 win at 10-man Karlstad, as Halmstad earned a second 1-0 win, this time against Varberg.
Group 4 is very close to call, as Goteborg won 4-0 at Trelleborg but they only lead Degerfors on goal difference, after they were held to a 1-1 draw by Ostersund, whose two points so far gives them a real chance of progressing.
Allsvenskan champions Mjallby have all but reached the last eight after winning 3-0 at Orgryte in Group 1, while cup holders Hacken also have a three-point lead in Group 7 as Adrian Svanback bagged a brace in a 4-1 win over Vasteras, who were unable to earn a repeat of their defeat of AIK last week.
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