The striker, who had not scored for his adopted homeland since defecting from Brazil in February, netted Spain's third goal in a comfortable success on Sunday.
Having lost 2-1 to Slovakia on Thursday, coach Vicente del Bosque named the youngest side of his six-year tenure, including goalkeeper David de Gea stepping in for out-of-form captain Iker Casillas.
Juan Bernat capped his international debut with a late strike to add gloss to the scoreline for the much-changed European Championship holders.
Silva showed typical composure and poise to bring Chris Philipps's poor clearance under control before lashing a superb half-volley into the top corner from 25 yards.
Costa's search for a first Spain goal continued to be a frustrating one though as he wasted a gilt-edged opportunity on the half-hour mark by skewing woefully wide from eight yards with only goalkeeper Jeremy Joubert to beat.
Alcacer displayed similar profligacy, somehow shooting off-target at the near-post, while De Gea had to remain alert to tip Daniel Da Mota's cross-cum-shot over at the other end.
However, Spain furthered their grip on the game two minutes later as Alcacer atoned for his earlier miss by converting Silva's reverse pass with a fierce shot into Joubert's right-hand corner.
Spain took their foot off the gas in a second half of few opportunities but Costa's moment arrived in the 69th minute.
The former Atletico Madrid man lashed home from point-blank range following a goal-mouth scramble, ending his dry spell at 516 minutes of football with a strike he will hope can kick-start his career for the Iberian nation.
And substitute Bernat took advantage of a weary Luxembourg side to add a fourth in the 88th minute, toe-poking into the bottom corner to add positivity to Del Bosque's apparent new era
