Tim Melia, who would be replaced by Kempin in the second half after suffering a head injury, prevented Kansas from falling just behind in the fifth minute when he kept out Liam Ridgewell's header, but the early pressure was a sign of how the first half would play out.
It was the Portland defenders who continued to look dangerous at set pieces as Nat Borchers lost his marker at a corner to head just wide of the post.
Kansas responded well to the Portland pressure before the break, and they started to cause more problems for the home defence as Dom Dwyer fired into the side-netting after linking up with Krisztian Nemeth early in the second half.
Darlington Nagbe came close to breaking the deadlock at the other end when his shot drifted just a few inches past the post, but the Timbers were celebrating the opening goal just moments later.
Diego Valeri fought his way into the box after Kansas had failed to fully clear the danger from a corner, and Wallace capitalised when the ball broke to him by picking out the bottom corner with a fine finish.
Portland appeared to be easing to the win as they held off the Kansas attack for long spells, but a twist arrived with just three minutes of normal time remaining.
Graham Zusi provided the perfect cross as Ellis pulled into space to head home the equaliser and send the game into extra time.
The winner appeared to arrive in the sixth minute of extra time as Nemeth received the ball close to the byline, before creating a yard of space and leaving Kwarasey helpless by somehow finding the top corner from a tight angle.
However, there was still time for more drama with just two minutes remaining when Manuel Asprilla found the space to deliver a cross which was impressively converted by Urruti.
Diego Valeri's opening penalty was kept out by Kempin, but Dwyer then saw Kwarasey stop his effort as Portland drew level after four spot kicks.
The drama continued as Jewsbury and Matt Besler lifted their penalties over the crossbar, before Urruti piled the pressure on Paulo Nagamura, who sent the game to sudden death by finding the bottom corner.
Ellis had the chance to secure the win after Kempin had denied Jorge Villafana, but the defender could only strike the outside of the post.
With the shootout coming down to the goalkeepers, Kwarasey held his nerve from the spot before getting two hands to Kempin's penalty to finally secure a memorable triumph.