United, who visit Home Park in the league on Saturday, put on a strong performance in Devon, with the only surprise being their failure to secure deserved promotion to round five in regular time.
Extra-time goals from Georginio Rutter and Crysencio Summerville, as well as an own goal by Ryan Hardie, settled the FA Cup fourth-round replay and secured Leeds’ place in the last 16, where they will face Aston Villa or Chelsea.
Brendan Galloway’s 78th-minute goal for Argyle cancelled out Wilfried Gnonto’s 66th-minute opener, forcing extra time, with Leeds leading for much the entire second period.
Farke, whose side hosts Yorkshire rivals Rotherham United in the league this weekend, said: “It was almost the perfect away performance. We were dominant, creating numerous opportunities and scoring goals, but we also struck the crossbar twice and the post.
“The scoreline favored us, but we had to play 120 minutes and it was a rerun.
“First and foremost, congratulations to Plymouth; they are a brave and confident team that never knows when they are beaten.
“Despite our dominance in both games, they found a way in.
“We dominated the second half; I just saw the stats, and we had 82% possession. But Plymouth had the free kick and scored, so they were fairly effective.
“Plymouth scored from nothing, but we showed amazing heart to win this. We put so much energy and spirit into the game and demonstrated a genuine desire to win.
“We took risks, pressed them high, and defended with discipline. I am pleased with our performance since we provided everything.
“I don’t plan ahead, but we may think about the next round when it comes around. “It’ll be an exciting game.”
Argyle manager Ian Foster said: “I told them after the game how proud I am of them.
“We talked about leaving with no regrets, and I’m proud of how we approached the game and made it difficult for Leeds throughout two games.
“The final scoreline is sobering for us because of their brutality. We won’t be disappointed with the outcome since we know how well we performed across two games.
“Look at the quality that Leeds had on the field late on; it makes it difficult for you.” We will learn from competing against high-quality opponents.
“Leeds are a fantastic team that deserved to win on the balance of the game, and we wish them the best.
“We’ve been caught a few times, which is OK. We will stick together and continue to learn and improve. We are a young, inexperienced team at this level.
“It’s quite difficult because it was a sobering scoreline but we will take the positives from it, dust ourselves down and move on.”
