While most La Liga clubs fielded much-changed teams against lower level sides, Atletico coach Diego Simeone selected several regulars in his side for the game on the outskirts of Barcelona.
But even with big names like Conor Gallagher, Alexander Sorloth and Rodrigo Riquelme in the starting 11, Atletico struggled on a soaked artificial pitch.
Simeone had to bring on Antoine Griezmann, captain Koke and Alvarez to help break the deadlock in the second half.
After the home side's Marti Riera wasted two golden chances on the counter, Atletico took the lead with a penalty in the 81st minute converted by Alvarez after Giuliano Simeone was brought down.
Vic's Alfons Senye was sent off for complaining to the referee following the penalty award and Alvarez wrapped Atletico's win after a counter attack in the 89th.
Atletico joined Girona, Villarreal, Rayo Vallecano, Espanyol, Sevilla, Celta Vigo and Getafe in the second round, with all La Liga sides advancing so far.
Seven Copa del Rey matches scheduled for Wednesday and Thursday were postponed after flash floods swept the eastern region of Valencia, killing at least 158 people.
The games featuring Valencia, Levante, Manises, Jove Espanol and Hercules de Alicante have been rescheduled after authorities advised against all non-essential travel in the region.
The floods in the southern region of Andalucia also led to the suspension of Cup matches featuring Chiclana and Xerez.
Six of the postponed games have been rescheduled for next week, leaving Real Sociedad's match against Jove Espanyol in Alicante as the only tie without a new fixture date.
Spain's FA announced earlier on Thursday the postponement of all games scheduled for Valencia's eastern region this weekend.
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