Top plays bottom at the Bernabeu on Saturday night, with the hosts reeling from their 4-3 aggregate quarter-final cup defeat to fierce rivals Barcelona - which came despite battling from 2-0 down to draw 2-2 on Wednesday night at the Nou Camp.
But Alonso does not think Jose Mourinho's team should be disheartened as they turn their attention back to La Liga, where they lead Pep Guardiola's side by five points at the season's midway stage.
He told www.realmadrid.com: "We gave a very good performance. It's a shame really, but feeling we were superior is important.
"We had to score three goals and we didn't manage it, but we fought to the end. We knew it would be difficult but we tried to the very end.
"The team showcased psychological strength. We never gave up and continued to fight. It just didn't happen.
"Our two goals may not have been enough to win the tie, but we're proud because we were better than them. We may have lacked a little luck. We had more chances to score than them throughout the match."
Madrid will be without Sami Khedira and Angel di Maria through injury on Saturday as they look to pick up where they left off against Zaragoza in their first meeting - winning 6-0 at La Romareda on the opening weekend of the season thanks to a Cristiano Ronaldo hat-trick and further goals from Marcelo, Xabi Alonso and Kaka.
Last weekend's goalless draw at Levante extended Zaragoza's run without a win in La Liga to 12 games, with the Aragonese side now seven points adrift of safety.
Loan signing Apono has urged his new team not to be overawed by the capital club, though - and firmly believes they can spring a surprise.
The midfielder, who made his debut at the Ciutat de Valencia after joining on loan from Malaga for the rest of the season, told Marca: "Sure, we can cause an upset. We have nothing to lose and everything to gain.
"We are last in the standings, it is true, and Real Madrid are in a different league, but we believe there is much to gain. Any positive result would be good for team morale.
"We have to respect them but not fear them. We are playing against the leaders and we know that, man for man, they have more quality than us.
"What you need to do is know what we are doing at the moment. The idea is very clear. If you do not focus, it's almost certain they score, so we have to be vigilant and take our chances.
"I think it's possible to win there. The situation is complicated because Real Madrid are a great team and we're down at the bottom. But doing things well, not being nervous and knowing what we can achieve... we can do them much harm."
On the overall challenge of staying in the top flight, the 27-year-old added: "I know where I have come. The situation is complicated, but I have come to help. I like challenges and hopefully we can save ourselves.
"The team continues to believe and the only thing missing is results."