Barcelona laboured for much of the match at the Pedro Escartin stadium, holding over 80% of possession but struggling to break down a disciplined five-man Guadalajara defence.
The home side posed a constant threat on the counter-attack and frustrated the European heavyweights for long stretches.
However, in the 77th minute, Frenkie de Jong whipped in a cross from the right and defender Christensen rose highest to meet the delivery, with his header deflecting off Guadalajara's Julio Martinez before nestling in the net.
Guadalajara's hopes were extinguished in the dying moments when Lamine Yamal played a sublime through-ball into the path of Rashford. The English forward showed great composure, wrong-footing keeper Dani Vicente with a quick cut to his left before tapping the ball into the empty net to wrap up the result for Barcelona.