HJK were hoping to make home advantage count in the second leg after a goalless draw in Bulgaria a week ago, even if they were not required to score five to come from behind as they were in the First Qualifying Round against NSI Runavik.
However, coming from behind would not be avoidable, as Arda took the lead midway through the first half when one casual pass at the back too many was intercepted by David Idowu, who drove inside the box and squeezed a shot from a tight angle past Thijmen Nijhuis.
But the Helsinki side responded well and did indeed level things up 13 minutes later - this time it was sloppy Arda passing in midfield that was pounced upon by Miska Ylitolva; he found Alex Ring who in turn picked out Kai Meriluoto, and he guided the ball into the bottom corner.
Extra time already looked to be on the cards in the early stages of the second half, as neither side registered an effort on goal as the tension continued to grow at Bolt Arena.
That all changed with just over 20 minutes to play, when a clearance was collected by Jere Kallinen, who slotted a clever ball through to Ring. His effort was thwarted, but Teemu Pukki was on hand to pick up the pieces, evade the defence and power his strike past Anatoli Gospodinov.
With seven minutes to play, Nijhuis made an excellent save to deny Idowu a second with a close-range header, before his defence held firm in the ensuing goalmout scramble.
But the home side could not hold out until full-time, as three minutes later Arda put together a flowing team move finished off by Birsent Karagaren, who received a pass from Georgi Nikolov and caught Nijhuis out at his near post with an effort from the left of the penalty box.
In extra-time, HJK were pushing for a winner, but despite the best efforts of Pukki and substitute Benji Michel, were struggling to find a way through and the match went to penalties.
Pukki got HJK off to a good start in the shootout, before 39 responded for Arda. However, Michel stepped up next and had his effort saved by Gospodinov.
The next four penalties were all scored, giving Idowu the chance to put his side 4-3 ahead with one shot each remaining, but he cannoned his strike against the crossbar.
But HJK failed to take advantage and put the pressure on their opponents, as Giorgos Antzoulas also fell victim to Gospodinov's heroics.
That allowed Andre Shinyashiki to dispatch the winning penalty for Arda, who have reached the Third Qualifying Round of a European competition for the first time.
Arda are two ties away from the UECL proper, and next they will face either Lithuanian side Kauno Zalgiris or Valur of Iceland. The first leg is due to be played next Thursday.