Spurs have not claimed silverware since they beat Chelsea in the 2008 League Cup final and they were made to work hard before seeing off the Championship promotion chasers in this one-leg semi-final at Hotspur Stadium.
Mourinho has won the cup on four occasions and goals from Moussa Sissoko and Son Heung-min ensured he now has the chance to make that five, having become the third manager to reach the final with three different teams.
"I came to England in 2004 and I remember that in that period I had to learn the meaning of the cups here and I always took it seriously," he said.
"If there is any secret it's to take it seriously. To respect what English football is, what English clubs are, what English lower divisions are.
"That's what I hope to do on Sunday in Crosby against Marine [in the FA Cup]."
Brentford had beaten four Premier League sides in a run to their first major semi-final but they were behind after only 12 minutes when Sissoko headed Sergio Reguilon's cross beyond keeper David Raya.
Thomas Frank's side thought they had equalised after the hour when top scorer Ivan Toney headed in but the goal was ruled out for the most marginal of offsides when the striker was actually down on his haunches earlier in the move.
Spurs increased Brentford's sense of injustice when Son raced clear to add a second after 70 minutes before the visitors were reduced to 10 men following Josh Dasilva's foul on Pierre-Emile Hojbjerg.
Mourinho's side will now meet either Manchester United or Manchester City the final at Wembley in April.
"If you see the winners you realise the big clubs want to win it," Mourinho said.
"Who was the last winner apart from the normal top six? Swansea? I remember Manchester City winning a lot of times, Chelsea winning a lot of times, I remember United winning a few times.
"I remember Liverpool in finals, Arsenal. The big clubs care about it."