The visitors started the tie with a one-goal lead to defend, having picked up a narrow victory in the first leg last week, and they seemingly planned to protect that lead rather than chase an opener early on.
Panathinaikos started positively and were allowed plenty of possession but could not fashion any clear-cut chances in the opening 20 minutes.
Having weathered an early storm, Vikingur grew into the game and had the first big chance of the match when through on goal, Ari Sigurpalsson rounded the goalkeeper before seeing his effort cleared off the line by a heroic Sverrir Ingi Ingason block.
After Valdimar Ingimundarson sent an effort narrowly wide, the hosts finally created some openings just before the break, the best of which fell to Nemanja Maksimovic, who pounced on a failed Vikingur clearance to send a snapshot just wide of the post.
With so much on the line, the second half unsurprisingly started at a frantic pace with chances for Panathinaikos coming thick and fast. The first of these fell to Fotis Ioannidis, who finally forced Ingvar Jonsson into a good save with a low strike. This was then followed up by a Karol Swiderski effort that was also smartly thwarted by Jonsson.
The hosts' pressure finally paid off 20 minutes from time, and it was more poor defending that put Vikingur in trouble.
This time, after a passage of pinball in the visitors' penalty area, the ball fell to Filip Mladenovic, who did brilliantly to find enough space in the area to bend a wonderful effort into the top corner, putting his side back in the tie with the fifth European goal of his career.
Vikingur barely got out of their own half in the closing stages of the second half, but a golden opportunity to edge ahead arrived in stoppage time when Sveinn Thorkelsson tested Bartlomiej Dragowski with a header from a corner.
Thorkelsson was made to pay for that missed opportunity in the cruellest of fashion when moments later, Tete latched onto Filip Duricic's rebounded effort to score a last-gasp winner and send Panathinaikos into the last 16 of the Conference League.
Flashscore Man of the Match: Tete (Panathinaikos)