Leeds were staring at a third-round exit for the sixth successive season until 18-year-old Sonny Perkins rescued them with his first goal for the club in the third minute of stoppage time.
Cardiff had led 2-0 through Jaden Philogene and Sheyi Ojo, and the Leeds players were booed off at half-time by a large travelling support.
But, inspired by a triple substitution just before the hour mark, Leeds rallied and Rodrigo headed home after 65 minutes.
The Spanish substitute then saw his penalty saved after Bluebirds defender Joel Bagan had handled and was sent off before Perkins squeezed home from close range to secure a replay.
“The whole first half we were not in sync and not aggressive enough,” American Marsch said of his maiden FA Cup tie.
“We didn’t have a full understanding of what this tournament and this game was going to require.
“We need better tactical understanding and savviness and awareness for exactly what matches require.”
The fourth-round draw was made within moments of the final whistle in the Welsh capital, with Cardiff or Leeds visiting the winners of the Boreham Wood or Accrington replay.
If Leeds progress, it will be the 11th consecutive time they have been handed an away tie in the FA Cup draw – equalling the competition record set by Stockport between 1996 and 2001.
Former Red Bull Salzburg and Leipzig boss Marsch said: “We know there’s an opportunity back at Elland Road with the draw, and everything is laid out in front of us.
“If we learned one thing is that if you don’t show up and you’re not ready to play from the beginning, you’re going to cause yourself problems.”
“I’ve never lost as a manager to a lower division team in the cup.
“Part of it is that I demand the players take it seriously and show the intensity as if it’s a Premier League match.
“We didn’t do that in the first half and we put ourselves in jeopardy. We have to learn now with a young group what cup matches are and how we handle it.
“Part of the challenge is also our team to teach them to be bigger winners and manage moments, what real games are about.
“But we also show heart and we will use this as a stepping stone to move forward.”
Cardiff, 20th in the Sky Bet Championship, have not won since beating Sunderland 1-0 away on November 5.
The Bluebirds are the lowest scorers in the Championship, but this was a much-improved display from Mark Hudson’s side.
Hudson said: “First and foremost, the overriding feeling is I’m proud of the performance we put in against a top Leeds side.
“Tactically, we got it right. We showed a good shape. We countered when we could, we frustrated them. We scored two really good goals.
“We knew we would suffer in the second half. They have got an abundance of talent to come off the bench.
“It’s not bad when you can bring Rodrigo off the bench.”