HJK were completely outclassed in the first leg a week ago, but their hopes of at least mounting a fightback were given a boost when Runavik's Fabian Ness was shown a straight red card after 15 minutes.
It took just three minutes for the home side to make the man advantage count, when Lucas Lingman curled the ball into the top corner from 25 yards out.
The Helsinki side were made to wait until the early stages of the second half to pull another goal back - this time Lingman was the provider, as his corner was flicked home by the head of Alex Ring.
The grandstand finish the Bolt Arena was hoping for was set up 11 minutes from time, when after patient build-up play, Santeri Hostikka tried his luck from outside the area.
What should have been a routine saved was spilt by Kristian Joensen in the Runavik goal, and the ball nestled in the bottom corner for 3-0 on the night and 3-4 on aggregate. Yet, it looked as though the Finnish side were going to come up just short of an incredible comeback.
That was until the 94th minute - after Ring's corner was cleared back out to him, his second attempt to pick out a teammate proved more successful, finding Giorgos Antzoulas who planted his header into the back of the net as pandemonium ensued all around him.
A 4-4 aggregate score sent the tie to extra time, and with all the momentum behind them, HJK went ahead for the first time just three minutes in through Hostikka's second of the game, this time curling a free-kick beyond Joensen and into the top corner.
HJK had six efforts on target in the extra 30 minutes alone against opponents who were running on empty, but the Hostikka strike was enough to complete a remarkable 5-0 win on the night and a 5-4 victory on aggregate.
In the Second Qualifying Round, HJK will take on Bulgarian side Arda.
There would be no miracle comeback for Ilves in the First Qualifying Round of the UEFA Europa League (UEL), but they at least restored some pride by drawing 0-0 with Shakhtar Donetsk, who prevailed 6-0 on aggregate following their impressive first leg showing.
Shakhtar arrived in Tampere looking to increase their lead - Pedrinho had an effort fly just wide early on, before Newerton saw a strike come back off the post.
There was more of the same early in the second half, when Lassana Traore perhaps should have found the target, then Newerton was denied again after a block on the line in a crowded penalty area.
In total, the Ukrainian side had 17 shots to Ilves' two over the 90 minutes, but the Lynx had done enough to save face as their UEL campaign came to an end, but they remain in Europe for 2025/26, dropping down to the UECL, where they will face AZ Alkmaar in the Second Qualifying Round.
Like HJK, SJK also needed a comeback against Faroese opposition in the UECL, but after a 2-1 defeat at home to Klaksvik last week, they were beaten 1-0 away in tonight's second leg.
SJK forced a few efforts on goal in the first 45 minutes but could not force them home, and that lack of cutting edge was punished just after the hour mark when Arni Frederiksberg put Klaksvik ahead.
The match and the tie were put to bed 12 minutes later, when Frederiksberg turned provider for Pall Klettskard, and the 35-year-old made it 2-0 on the night and 4-1 on aggregate.
