The Faroe Islands were able to break the deadlock shortly before the interval when Christian Holst smashed the ball in from the edge of the box against the run of play.
Finland were back on level terms early in the second half when Riski hit the ball low on a rebound from a Joan Edmundsson effort moments earlier.
The visitors continued to create the majority of the chances, but Nielsen pulled off a series of saves to keep the Faroe Islands level for much of the second half.
However, Riski was on hand to pounce again with just over 10 minutes to play when Nielsen parried from a Pukki shot, only to direct the ball into the path of the goalscorer.
Finland put the win beyond doubt four minutes later as Roman Eremenko burst forward from midfield before creating some space around 25 yards out to unleash a thunderous strike.