Steven Naismith atoned for a poor first-half miss to play a prominent part in both goals and give Alex McLeish his second victory since returning to the post in February.
The Hearts striker will contest the credit for Berat Djimsiti's own-goal opener in the 47th minute but there was no doubt over the second he later headed home to cap his country's confident response to last week's 4-0 humbling against Belgium.
His goals mean McLeish's men have ended a run of three matches without scoring on home soil and, more importantly, join their opponents on three points at the top of Group 1 in League C.
Scotland, evidently stimulated by its Belgium embarrassment, was full of endeavour in the early exchanges, swarming forward to Albania's consistent anxiety.
All that was missing was the reward for effort Naismith remarkably failed to provide when he nodded against the post at point-blank range in the 24th minute, Charlie Mulgrew having climbed to knock down a free-kick.
Redemption initially came with a slice of fortune as the assistant referee failed to spot the 31 year-old straying marginally offside at the far post before heading substitute Stuart Armstrong's cross in off the unknowing Djimsiti.
Bekim Balaj squandered two opportunities to restore parity but, unlike for his fellow striker, there was to be no reprieve as Naismith instead sealed the result with 22 minutes to play, punishing Thomas Strakosha for flapping at a routine corner.