The Irish Premiership champions suffered a disastrous start in the Faroe Islands when they conceded a goal with less than three minutes on the clock.
Vikingur broke quickly and a low cross from the right from Solvi Vatnhamar fell into a dangerous area between Hall and goalkeeper Chris Johns and the central defender, at full stretch, could only divert the ball into his own net.
Linfield were in arrears for only six minutes, though. Chris Shields made strides down the right-hand side and his cross was met with a glancing header by Fitzpatrick and the striker's effort had enough power to loop over the outstretched arms of goalkeeper Hans Jakup Armgrimsson.
Buoyed by that goal, Kieran Offord tried his luck from 25 yards soon afterwards, but Armgrimsson was alert to turn the effort over the crossbar.
The Scottish striker was again denied by the goalkeeper on the half-hour mark after bursting into the penalty area, but his fierce shot was palmed away at the near post.
The home side had a rare sight of goal just before the interval when Vatnhamar's corner was headed goalwards by defender Ari Olsen, but Johns was able to claw the ball away for a corner.
Kirk Millar saw a 25-yard strike tipped over before Linfield suffered a hammer blow on 55 minutes.
There seemed little danger when left-back Jonhardsson's cross from the left was cleared back out to him, but the full-back collected the loose ball and drifted infield before arrowing a low right-footed shot into the bottom corner beyond the reach of Johns.
Despite pushing forward in search of an equaliser, Linfield's attacking efforts after the break lacked the fluidity of the first period and the home side defended stubbornly to preserve their lead which they will take to Belfast.
