Pique headed the Catalans in front in the 36th minute, connecting with a cross from the lively Ousmane Dembele who produced another decisive display after scoring Barca's late equaliser against Atletico Madrid last week.
Villarreal are hovering above the relegation zone but they produced an admirable display and nearly took the lead when Gerard Moreno seized on a mix-up between goalkeeper Marc-Andre ter Stegen and defender Clement Lenglet but was able only to strike the outside of the near post.
Homegrown midfielder Alena, 20, came off the bench to score his first league goal for Barca in the 87th minute and seal the win, racing on to a through ball from Lionel Messi and producing a confident finish.
"We completely dominated the play in the first half despite them hitting the post but we lacked a bit of dynamism in the area but we were rewarded with Pique's goal," Barca coach Ernesto Valverde told reporters.
"In the second half, we lacked the ball and when that happens, we suffer a lot. We gave little away because we were organised but we do not feel comfortable playing that way."
Barca lead the standings with 28 points after 14 games, while Sevilla are second on 27 after drawing 1-1 at Alaves. Atletico Madrid are third on 25 after being held to a 1-1 draw at Girona.
Barca had dropped off the top of the table following last week's 1-1 stalemate at Atletico in which they had struggled for creativity.
They lacked a spark against Villarreal too, badly missing the control that injured midfielder Arthur Melo has provided in recent weeks.
Record signing Philippe Coutinho struggled to make an impact and was booed by an impatient Nou Camp crowd when he was substituted in the second half, while fans also showed their discontent when Valverde took off Arturo Vidal for Alena.
Alena had made only one league appearance this season after being promoted from the club's reserve team but he eventually lifted the mood by producing the lethal blow to Villarreal, who had deprived Barca of the ball for much of the second period while rarely troubling Ter Stegen.
(Reporting by Richard Martin; editing by Clare Fallon and Ian Chadband)
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