Argentina forward Angel di Maria ran onto a perfectly-timed pass from fellow substitute Karim Benzema to open the scoring in the 72nd minute. Alcoyano's Javi Selvas was sent off a minute later for arguing with the referee about an offside decision, earning him his second yellow card of the game.
Jose Callejon then followed up with a goal in the 89th minute and another in injury time.
The Santiago Bernabeu meanwhile welcomed winger Denis Cheryshev, who became the first Russian to start a match for Madrid. He was replaced in the 61st minute by Di Maria.
"I'm very grateful to the coach and above all happy with how the other players and the fans have received me," Cheryshev said.
Mesut Oezil, who has been criticized recently by coach Jose Mourinho for a lack of commitment and stamina, struggled to find a way through Alcoyano's defence.
However, he proved decisive in supplying Callejon with a beautifully-timed pass for his first goal.
In other round of 32 matches, Real Betis, Las Palmas, Cordoba, Osasuna and Malaga also went through.
Betis beat Valladolid 3-0 and advanced 3-1 on aggregate. Antonio Amaya opened the scoring for Betis, heading home a corner in the 27th minute, before Alvaro Vadillo beat two defenders and played a quick one-two with Ruben Castro to score in the 63rd minute.
A hapless Jesus Rueda scored an own goal in the 87th as Seville followed up its 1-0 win over Real Madrid 1-0 in the league at the weekend.
Cordoba went through 4-2 on aggregate after settling for a 2-2 draw with Real Sociedad.
Antoine Griezmann put Sociedad ahead in the 20th minute of a high-tempo match. Cordoba's Kiko Olivas cut through to score in the 65th before teammate Sebastian Dubarbier made it 2-1 to the Andalucian side -- only for Sociedad's Imanol Agirretxe to tap in the equalizer in the 85th minute.
Osasuna beat Gijon 2-0 for a 2-1 win on aggregate. Osasuna's Joseba Llorente scored in the 71st minute and Ruben latched onto a poor clearance six minutes later to double his side's lead with a delicate side-footed tap into the net.
Malaga lost 1-0 at home to Cacereno but made it through on the away goals rule after a 4-4 draw on aggregate.
The goal was a controversial one with Alberto "Chapi" Macias seeming to collect the ball in an offside position before scoring in the 38th minute.
On Wednesday, it's: Valencia vs. Llagostera; Atletico Madrid vs. Real Jaen; Levante vs. Melilla; Getafe vs. Ponferradina; and Espanyol vs. Sevilla.
On Thursday, it's Celta vs. Almeria; Granada vs. Zaragoza; Mallorca vs. Deportivo; and Athletic Bilbao vs. Eibar.
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