Tuesday night's game was delayed because of problems with the stadium's temporary grandstand.
Kick-off was held up for half an hour because local authorities did not initially allow fans access to the newly built grandstand at the Pedro Escartin stadium in the city of Guadalajara, near Madrid.
There were long lines outside until police finally permitted supporters inside. The stadium's capacity increased from nearly 6000 to about 8000 with the new stand.
Barcelona were in control from the start but struggled to create significant scoring chances against their lower-tier opponents.
Danish defender Christensen finally broke the deadlock in the 76th minute with a deflected header off a cross by Frenkie De Jong.
The hosts had a few opportunities to equalise but couldn't capitalise and England international Rashford sealed the victory on a breakaway in the 90th.
Barcelona coach Hansi Flick rested some of the team's regular starters ahead of Spanish league leader's visit to third-placed Villarreal on Sunday.
"These are tricky matches; they come out with everything they've got and we have to do the same," Barca defender Pau Cubarsi told TVE.
"In the second half, we picked up the pace and moved them from side to side to tire them out. As the minutes passed, they grew tired, and �that was the key point. They played spectacularly well."
In the upset of the night, second-division Deportivo La Coruna eliminated top-flight Mallorca 1-0.
The other first-division clubs advanced. Real Sociedad defeated Eldense 2-1 with a winner deep into added time, while Valencia beat Sporting Gijon 2-0.
On Wednesday, Real Madrid visit third-division club Talavera, while Atletico Madrid are at fourth-tier Atletico Baleares.
With Reuters.
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