In Pygmalion, how can we justify that emotions evoke knowledge?
That's an excellent, if unseen, aspect of Pygmalion. I believe you're asking how can we validate that an awakening of feeling can lead to an awakening of new found thought. It's actually quite a strong point if one looks at the characters of Eliza and Higgins. Take the moment when Higgins and Pickering return from their party flushed with happiness over their success at winning the bet. They completely ignore Eliza who has learned more about herself over the last few weeks than she has in her entire life. No longer is she making incomprehensible sounds or stalking out of the room when she is upset at the way she is treated (as has happened in the past). Anger at the two men slowly boils within her gaining strength. She realizes that for all her fine clothes, speech and mannerisms the fact that she was born in the poorer section to a common laborer will forever separate her from those with whom she has hobnobbed with until now. As a flower girl she believed that mimicking her "betters" was enough for a better life. Now, despite all she has learned, she is still a "squashed cabbage leaf" to Higgins. He will still ask her to fetch and carry; he wouldn't ask the same of the Eynsford-Hills, despite his contempt of them. She sees this with perfect clarity; she tells Higgins, "There can't be any feelings between the like of you and the like of me." There is a wall between them that she always thought was one of money and accent but has now learned differently.
Higgins has also come to a new realization when Eliza walks out the door. He thought he would never need anyone, except for convenience like Mrs. Pearce. Instead he's outraged by the audacity Eliza displays when she coolly tells him she no longer needs him. She has learned everything she needs to become a lady, he is unnecessary. His anger at hearing that Eliza may marry Freddy stems from the fact that he believes she is throwing away a mind and character worthy of challenging him on an imbecile. He realizes that she is his equal and match. He doesn't have the maturity to ask for more than a squabbling partner so he loses her but he learns something new about his own character.