Wisla, already beaten by Odense in their opening Group K clash, took a shock ninth-minute lead through David Biton, but Janko's double and goals for Luuk de Jong and Willem Janssen eased Twente home.
Twente might have been fresh from a fine Eredivisie draw at Ajax on Saturday but things did not go to plan in the opening minutes as Biton fired past Nikolay Mihailov from 20 yards to give his side the lead.
Wisla had entered the game on the back of three straight domestic wins so may have fancied their chances, but Twente hit back with Janko and Wout Brama both bringing saves out of Sergei Pareiko.
Denny Landzaat and Emir Bajrami also kept Pareiko busy as the hosts pushed for an equaliser, and it arrived in the 32nd minute when de Jong rose to head home a free-kick from Ola John.
Biton missed a rare chance for Wisla to regain the lead before Twente went in front in added time at the end of the first half when John fed Janko from the left and the strike shot home low into the bottom corner.
Wisla made a good start to the second half with Ivica Iliev firing wide then bringing a save out of Mihailov.
But Twente pushed again with Pareiko making a punching save to deny Bajrami before Janko grabbed his second on 57 minutes after finishing off a Peter Wisgerhof through-ball.
Radoslaw Sobolewski shot wide for Wisla before Janko went close to completing his hat-trick with a 63rd minute effort which clattered the legs of Pareiko.
Bajrami hit the bar with a curling left foot shot on 66 minutes, and 10 minutes later the same player hit the bar again with a powerful header on the end of a cross from John.
But Twente finally grabbed their fourth and rounded off their night in the style when Janssen blasted home from a Roberto Rosales cross with 10 minutes left on the clock.