In an "away" match at HJK's Bolt Arena, Klubi 04 held out until half-time thanks to Lucas Lingman missing a penalty 10 minutes before the break.
However, the second half was one-way traffic after Santeri Hostikka broke the deadlock five minutes after the restart, and just two minutes later, Lingman set up Liam Moller for HJK's second.
Benji Michel made it two in a week for him inside the final 20 minutes, before Moller grabbed his second to secure a 4-0 win and keep the Helsinki side on course for a 15th Suomen Cup.
Haka took league and cup holders KuPS to penalties in one of two all-Veikkausliiga Quarter-finals, only to miss all three of their spot kicks and hand the Kuopio side the win after their 1-1 draw.
The hosts fell behind in the 12th minute after a Saku Savolainen strike, but Mikael Almen drew Haka level just over 20 minutes later. That's how it stayed until the final whistle, sending the match to penalties.
Oskari Sallinen, Almen and Arlind Sejdiu all missed for Haka, while Ibrahim Cisse, Arttu Heinonen and Agon Sadiku all scored to give KuPS the 3-0 win and some welcome respite from their poor league form.
After Klubi 04's elimination, Jaro ensured there would be no lower-league representation in the Semi-finals, as they saw off a spirited Ekenas, 4-2 away from home.
Salomo Ojala scored twice for the Ykkosliiga side in the second half, but a brace for Rudi Vikstrom and one each for Jim Myrevik and Samouil Izountouemoi sent Jaro into the last four.
Finally, AC Oulu, bottom of the Veikkausliiga table, are into the Suomen Cup semis after a 2-0 home win over SJK.
Leon Bergsma broke the deadlock in the sixth minute, and it took until the 89th for Tuomas Kaukua to settle the tie.