The 27-year-old said that United had been beaten by the better side in Skopje and insisted that the club must dust themselves down and get ready to go again.
The Spaniard told MUTV: 'We have to accept Real deserve to be the winners. But we can take a lot of good things from this game.
'The Premier League starts on Sunday, there is no time to be upset, we want to be contenders for the title and we want to fight for every title this season.'
Despite last season's trophy haul, Herrera says that the club want to challenge for the big one, the Premier League title, this time around.
'Last season, we won three trophies which is very good, but we didn't fight for the Premier League.
'That's our aim for this season and we have a very good squad. We are going to play in four competitions and we want to fight for all of them.
'In every position, we have two very good players, so I am optimistic for this season and let's forget about tonight.
'I know it's difficult because it's a European trophy and that's important for the club.
'But now the Premier League, then the Champions League will come, the FA Cup and the League Cup. We don't want to give away any competition.
'We are going to fight for all of them and we are going to improve.'
After admitting that Real, who he described as the 'best team in the world', were just too good for United on Tuesday, the Basque insisted that United's focus was now completely on Sunday's Premier League opener against West Ham.
'We are ready and West Ham will be ready as well, because they have also played a pre-season and surely they are going to be fit as well,' he explained.
'But we are playing at Old Trafford in front of our fans and we want to win the first game of the season.
'We have to respect them because they have a very good team, they have signed very good players.
'This season, I think the transfer window has gone good for them. They are going to be difficult to beat. Now is our chance to fight for the Premier League and we have to do it as well.'