Nigeria offered nothing during a lacklustre first half but Musa turned on the style after the break, first finding the roof of the net with a brilliantly-taken half-volley before a fine individual effort doubled his side's advantage.
A point against Argentina could now be enough for Gernot Rohr's team to qualify from Group D while Iceland - who missed a penalty through Gylfi Sigurdsson during the latter stages - must beat Croatia to maintain any hope of progression.
After Argentina's defeat to Croatia on Thursday night, both teams were aware that three points would go a long way to securing a place in the last 16 and it was Iceland who looked to force the pace.
Sigurdsson saw a free kick tipped over the crossbar as early as the second minute before the Everton playmaker squandered a good opportunity on the edge of the box with his weak curler being easily dealt with by Francis Uzoho.
However, Iceland failed to build on their start and with Nigeria content to keep hold of the ball without showing much intent to force the issue in the final third, the remainder of the first half featured very little goalmouth action.
A temperature of 32C played its part but just as it appeared that the first half would peter to a close, Iceland found a second wind with Alfred Finnbogason failing to apply a decisive touch when meeting Sigurdsson's set piece from the right.
During two minutes of added-on time, Jon Dadi Bodvarsson also wasted an opportunity with the Reading forward heading wide of the far post after getting on the end of a corner delivered to the back post.
Nigeria had offered nothing during the opening 45 minutes but the Super Eagles quickly opened the scoring at the start of the second half, with Victor Moses bursting down the right and finding Musa, who took a touch before lashing a half-volley into the roof of the net from 12 yards.
Iceland were still offering plenty in the final third but Nigeria were looking dangerous on the counter-attack, and Wilfred Ndidi almost added a second with a 30-yard strike which was tipped over the crossbar.
Up the other end, Rurik Gislason curled off target from around 20 yards out and it quickly became apparent that Nigeria were beginning to control the game, with Leon Balogun almost adding a second with a glancing header which went over.
Musa then rattled the crossbar with a long-range strike but moments later, the out-of-favour Leicester City man netted a sublime second as he outpaced Karl Arnason and rounded Halldorsson before firing into an empty net.
Iceland looked a beaten team but, with 10 minutes left, they were awarded a penalty - thanks to VAR - after Tyronne Ebuehi brought down Finnbogason inside the penalty area.
However, Sigurdsson directed his spot kick well over the crossbar, and it left Nigeria in a position to comfortably see out the rest of the fixture with minimal fuss and leave their World Cup fate in their own hands.
NIGERIA (4-2-3-1): Uzoho; Omeruo, Troost-Ekong, Balogun, Idowu (Ebuehi 46'); Mikel, Ndidi; Moses, Etudo (Iwobi 91'), Musa; Iheanacho (Ighalo 85')
ICELAND (4-4-2): Halldorsson; Saevarsson, Arnason, R.Sigurdsson (Ingason 65'), Magnusson; Gislason, Gunnarsson (Skulason 87'), G Sigurdsson, Bjarnason; Finnbogason, Bodvarsson (Sigurdarson 71')