Wilde struck twice in the first half – in the 15th and 27th minutes – and completed his treble in the 47th minute before Matthew Williams made it four on the night, while captain Lukasz Cieslewicz scored a consolation goal for the Faroe Islands outfit at the death.
TNS now face a second qualifying first-leg clash against Hungarian’s Videoton next week as they look to move a step closer to the Champions League proper.
Northern Ireland’s Crusaders advanced on away goals after their 1-1 result against Levadia of Estonia at Le Coq Arena with Michael Carvill’s early opener making the difference despite Siim Luts’ equaliser midway through the first half.
Elsewhere, there was success for Gibraltar’s Lincoln, who saw off Santa Coloma 2-1, while Armenian side Pyunik made it through after a 2-1 (4-2 aggregate) win over Folgore.
In the Europa League, Kieran Waters netted a brace to propel Shamrock Rovers into the second qualifying round thanks to a 3-0 aggregate victory over Progres.
Following a goalless first leg in Luxembourg, David Webster put the League of Ireland side ahead midway through the first half at the Tallaght Stadium.
Waters added his side’s second just before the break and made sure of the victory after 57 minutes as Pat Fenlon’s Hoops progressed.