The Germans, who suffered a shock 2-1 loss to North Macedonia in March, took the lead in the fifth minute when Sane played in Serge Gnabry to score, and Antonio Rudiger added a second with a header from a free kick 20 minutes later.
Iceland had a number of decent chances but Sane killed the game off when he rifled the ball into the roof of the net after another flowing German attack 11 minutes into the second half, and Timo Werner added a late fourth.
A constant threat with his quick dribbling and precise passing, Sane effectively killed the game off when he rifled the ball into the roof of the net after another flowing German attack 11 minutes into the second half.
After a shaky start to qualifying including a shock 2-1 loss to North Macedonia in March, the Germans have rediscovered their scoring touch, but were still guilty of a few glaring misses.
Werner made amends by adding a late fourth that hit the left-hand post and bounced out before spinning back across the line to round off an encouraging team performance.
The Germans have taken a maximum nine points from their first three games under new coach Hansi Flick, scoring 12 goals and conceding none.
"We only had a few training sessions. Our goal was to win the three games, and we achieved that. We have taken a step forward," Flick said.
"We're not at 100% yet, but the team is on the right track. The guys are implementing things the way we imagine."
Germany top the group on 15 points, four ahead of Armenia. Romania are third on 10 points with four games left to play, with Iceland fifth on four points.