Lulea had held off Orebro in the eighth-finals to set up a repeat of the Champions Hockey League final, which also went to overtime, but crucially it was the visitors celebrating a 3-2 win this time, to take a 1-0 series lead.
The opening goal went to the reigning SHL champions just over eight minutes in, when Eetu Koivistoinen dug out a shot under pressure, which was diverted in off the skate of the unfortunate Linus Hogberg.
But high-sticking from Isac Hedqvist four minutes later was all the invitation Frolunda needed to draw level, as the reliable Filip Cederqvist finished off the move to equalise on the powerplay.
Hedqvist made amends with six minutes to play in the second period, as he put Lulea back in front after being set up by Oskari Laaksonen, but again the visitors' lead didn't last, as Noah Dower Nilsson scored his first goal since 24th January three minutes into the third period.
That sent the match to overtime, where a trip by Tom Nilsson after six-and-a-half minutes proved fatal for the Gothenburg side, as 40 seconds later Laaksonen was again the provider, this time for Anton Levtchi to maintain his record of scoring once in every game this postseason and win it, 3-2.
Regular season title winners Skelleftea were forced to come from behind against the lowest-ranked side still left in the competition, Malmo, but in the end had a more comfortable opening night, winning 4-1 at home to go ahead in the series.
Following a goalless opening period, the Redhawks took the lead six minutes into the second when Lauri Pajuniemi took possession behind the net and forced the puck past Linus Soderstrom on the stretch.
But the visitors' lead lasted all of 15 seconds, as Oliver Okuliar set up Jonathan Johnson for Skelleftea's first postseason goal, before they completed the turnaround on the powerplay late in the period, when Oscar Lindberg struck the bar and Mikkel Aagaared reacted quickest to the rebound.
The number one ranked side finally got some breathing space midway through the final period when Okuliar set up Emil Djuse, before the Slovak made it a hattrick of assists late on when he teed up Andreas Johnson for an empty-netter that sealed victory.
Game two in both series will be played on Wednesday 25th March, 24 hours after the other two quarter-finals get underway, as Rogle meet Farjestad and Vaxjo welcome Brynas.