They will now aim to take another step towards a first La Liga title in four years at home to Sevilla on Sunday.
Madrid crashed out of Champions League on penalties at the semi-final stage against Bayern Munich in midweek, but have the chance to put their misery behind them this weekend.
Victory over Sevilla at the Bernabeu could even be enough to see them dethrone Barcelona if the Spanish champions - reeling from the news that hugely-successful coach Pep Guardiola is to leave the club at the end of the season - suffer a shock defeat at Rayo Vallecano later in the day.
"This team has pride and is looking forward to playing against Sevilla," Karanka told today's pre-match press conference.
"We are close to achieving something that would makes us very happy and we will fight to the end for it.
"It is a great pleasure and a privilege to fight with this team because it is always ready to pull through and win titles when the going gets tough."
Karanka added: "Tomorrow's [Sunday] will be a complicated match because Sevilla are only two points shy of qualifying for a European competition next year and they've been preparing for the match the last couple of days in Madrid.
"Three important points will be at stake in a race that is coming to an end.
"We're only concerned about winning tomorrow."
Meanwhile, Karanka chose to be sparing in his praise when asked for his views on Guardiola's impending departure from the Nou Camp.
"The Spanish league has existed without Guardiola, Mourinho, Karanka... and it will continue to exist," he said.
Madrid hope to get over Wednesday night's heartache against Bayern by aiming for a record-breaking end to the season.
'Operation 100 points' is said to be the mission for Mourinho's side, who must win all four of their remaining games to become the first La Liga team ever to reach three figures.
Madrid's previous best tally was 96 two years ago, although that was still three points shy of Barca's 99.
Four wins would also see Madrid surpass by one the record of 31 victories in a league season both they and Barca managed in 2009-10.
They have already beaten the competition's goalscoring mark by netting 109 times, with Cristiano Ronaldo smashing his own record from last season by bagging 42.
The Portugal star's next target is the European Golden Boot best of 47 held by Romanian legend Dudu Georgescu.
Madrid looked tired against Bayern following their 'Clasico' exertions and Mourinho may elect to freshen up his side on Sunday.
Sevilla, meanwhile, will try to take advantage of any Champions League hangover in a bid to keep alive their faint hopes of qualifying for next season's competition.
Michel Gonzalez's men are in ninth place but only six points adrift of fourth spot in a congested table below Real and Barca.
Young defender Juan Cala, who is hoping for his first taste of the Bernabeu, told his club's official website, www.sevillafc.es: "We will have our two or three opportunities and, if we take them, we can win the game."