Louis Moult's double knocked Gers out of the competition and sent the Fir Park side into next month's final against Celtic.
Prior to Sunday's encounter at Hampden Park, Caixinha had told his players that "we only have two options - to win or win".
Caixinha told BBC Sport after the game: "I assume all the responsibility because the team was a shade of what it needs to be.
"The opponents played better, we let them play their own game. Our performance was really, really poor.
"Maybe I created a lot of expectation, I am always like that, maybe I created a sort of anxiety. I don't know why they under-performed, individually and collectively. It is up to me.
"We got a little better in the second half, we had a couple of chances to score, but after the second goal, we just disappeared as a team."
Up next for Rangers is Wednesday's home Scottish Premiership match against Kilmarnock.