Scotland netted in either half as the struggling Canaries were dragged further into the mire.
The Trinidad & Tobago striker saw his first-half goal cancelled out by Alan Lee's header moments later, but Scotland took his goal tally to 23 this season with a second-half penalty as Bryan Gunn's side's survival bid was dented.
Swansea made three changes from the side that were held against rivals Cardiff last time out in the South Wales derby.
Norwich, who were looking anxiously over their shoulders, also made a reshuffle in a move which saw Lee return to the side.
Swansea began the better side and Garry Monk was denied his first goal of the season when his shot rattled the crossbar.
Former Ipswich striker Lee fired inches wide to spurn a glorious chance in 26th minute. And he was made to rue the miss moments later when Scotland rifled home a fine strike from 25 yards out.
The home side's joy was short-lived when Ryan Bertrand's cross saw Lee outjump Ashley Williams to plant his looping header beyond Dorus de Vries in the 33rd minute.
Scotland, a constant thorn for the visitors throughout, saw a goal chalked off for offside before David Marshall pushed his low drive around the post seconds after the break.
But he wouldn't be denied when he struck home a 50th minute spot-kick. Scotland picked himself up after he had been tripped by Gary Doherty to send Marshall the wrong way.
Andrea Orlandi and Scotland fired wide as they missed glorious chances to kill off the visitors.
But Norwich couldn't muster an equaliser after pressing to the death. Lee twice came close before Darrell Russell crashed a shot into the sidenetting.