Jose Mourinho's side are preparing to take on the Red Devils at Old Trafford on Tuesday with the last-16 tie delicately balanced at 1-1 following the first leg at the Bernabeu last month.
In the intervening weeks since that game Mourinho's men have won all four of their fixtures, including successive victories over great rivals Barcelona.
Former Liverpool midfielder Alonso believes those two El Clasico wins have instilled a belief in the Real team and they are confident of progressing through to the quarter-finals.
"The victories over Barcelona have demonstrated that Real Madrid can cope with the big battles," said the 31-year-old. "If we can win at the Nou Camp we can win at Old Trafford.
"The match at Old Trafford is like a final for us. Our dream is to win the Champions League."
Alonso also feels that his side deserved more than the 1-1 draw from the first leg, but has played down the importance of United's away goal on the outcome of the tie.
"In the Bernabeu we were the superior side and deserved a better score - [David] De Gea was decisive in the match," he added. "[But] the match in the Bernabeu is history and now the round will be resolved in 90 minutes.
"The tie in Madrid has not changed much. We said that there will be goals between the two teams and normally the score at Old Trafford is not 0-0."