Ferran Torres was also on target for the Catalan giants, who moved back into second place to trail leaders Real Madrid by five points.
Barca continued to struggle in defence and Elche had chances to cause an upset, scoring through Rafa Mir, with the striker also twice hitting the woodwork.
Hansi Flick’s Barca have failed to keep a clean sheet in six consecutive league games, in the midst of an injury crisis.
The champions were missing Raphinha, Pedri, and Joan Garcia among others, but were able to bring on Robert Lewandowski and Dani Olmo following spells out of action.
Teenage star Yamal underwhelmed in last week’s defeat by Real Madrid after coming back from a groin issue, but looked more sprightly against Elche.
“We found our spirit and intensity,” Torres told DAZN.
“I think we’re finding last year’s Barca, from the first minute to last we gave everything, and we’ll keep doing that.”
Coached by Eder Sarabia, a former Barcelona assistant coach under Quique Setien, the ninth-placed visitors came to play and impressed despite defeat.
Barca took the lead after nine minutes when Alejandro Balde drove forward and fed Yamal in the area.
The winger took a touch to set himself and then whipped a clinical finish past former Barca goalkeeper Inaki Pena.
Flick said Yamal’s groin injury was still an issue that the youngster needs to work on.
“He has to train, have treatments, he has to do this and he did it, it’s a good step forward for him,” explained Flick.
“I cannot say it’s over, it’s coming and going like this and he has to manage it.”
Three minutes later Torres doubled the hosts’ advantage, tapping home after Fermin Lopez broke into space down the left and provided an inch-perfect cross.
Pena saved from Torres as Barca squandered some presentable chances to add to their lead.
Mir, who in October was indicted for sexual assault allegations dating back to 2024, pulled a goal back shortly before half-time.
Barca tried to catch the forward offside but he was in his own half when he was played in. Mir bent a shot past Ronald Araujo and Barca goalkeeper Wojciech Szczesny and in at the far post.
Torres might have scored again before the interval but Pena produced a superb save to tip his strike to safety.
‘Days like this’
Early in the second half Mir clipped the top of the crossbar with a curling effort as Elche battled well.
Rashford sealed Barca’s victory with a superb finish after Lopez picked him out with a cross, for his second league goal of the campaign.
“When you see his chances one or two goals more would be good… but we can be happy for him,” said Flick.
Rashford admitted he could have been even more effective in the final third.
“I could have had more assists or goals, sometimes the decision making to shoot or pass was wrong today, but I am going to have days like this,” he told the club’s streaming platform Barca One.
Veteran goalkeeper Szczesny pushed a Mir effort onto the post to save Barca sweating over their advantage in the final stages.
“We were ourselves and the fans are proud of how we played,” said Elche coach Sarabia.
“We know that it’s normal that they score goals against us, but we could have got some too.”
Elsewhere Real Betis rose to fifth with a convincing 3-0 home win over Mallorca, inspired by Brazilian winger Antony.
The former Manchester United forward struck twice and created the third for Ez Abde as the Andalusians moved three points behind Atletico in fourth.
“Here I am very happy, I am at home,” said Antony, who flopped at Old Trafford but enjoyed a revival on loan at Betis last season before making the switch permanent.
“It was very important to win today to be up there (in the fight for the top four), we have to be patient and go game by game.”
