Hansi Flick's men flew out of the blocks, with Lamine Yamal firing home inside nine minutes and Ferran Torres doubling the advantage shortly after.
Rafa Mir pulled one back for Elche before the break, but Marcus Rashford's thunderous second-half strike restored Barca's cushion.
Despite a nervy finale, the Catalans held firm to climb back to second place, five points behind leaders Real Madrid. Flick also welcomed back Dani Olmo and Robert Lewandowski from injury ahead of Wednesday's Champions League tie against Club Brugge.
Elsewhere, Real Betis continued their fine form with a 3-1 victory over struggling Real Sociedad, moving Manuel Pellegrini's side into the top four.
Cucho Hernandez's early rocket set the tone, but Betis were pegged back by Brais Mendez.
A comical own goal from Sociedad keeper Alex Remiro restored Betis's lead before Pablo Fornals sealed the win.
Betis now turn their focus to a Europa League meeting with Nottingham Forest, while Sociedad are still searching for their first league win of the season.
In Vitoria, RCD Espanyol suffered a narrow 2-1 defeat to Deportivo Alaves.
Goals from Denis Suarez and Lucas Boye put the hosts in control before Roberto Fernandez struck to give Espanyol hope.
Despite a late surge, the visitors couldn't find an equaliser, leaving Mendizorroza with only frustration to show for a spirited effort.
Meanwhile in Valencia, Celta Vigo claimed a dramatic late 2-1 win over Levante UD.
Oscar Mingueza opened the scoring just before half-time against ten-man Levante, who had Unai Vencedor sent off.
The hosts leveled through Kervin Arriaga, but substitute Miguel Roman marked his first-team breakthrough with a stoppage-time winner to give Celta only their second league victory of the campaign.
Barcelona manager Hansi Flick has challenged Marcus Rashford to be more clinical in front of goal despite the English forward scoring in the club's 3-1 La Liga win over Elche on Sunday.
Real Madrid moved seven points clear at the top of LaLiga after an impressive 4-0 win over Valencia at Santiago Bernabeu on Saturday.
Lamine Yamal is determined to silence his critics and return to his best after a difficult showing in Barcelona's 2–1 defeat to Real Madrid in last weekend's El Clasico.
Barcelona's growing injury list has taken another major hit, with midfielder Pedri Gonzalez ruled out for several weeks after suffering a muscle tear in his left hamstring during the 2–1 defeat to Real Madrid last weekend.
