But their willingness to run with the ball was punished when Lee Blackett intercepted a loose pass on 70 minutes and sent Goodridge in for a try, converted by Thomas.
It was a different story at the start of the game when Chris Malone kicked two penalties to Thomas's one, but Leeds levelled again after 21 minutes from the boot of Thomas.
The visitors were then under pressure close to their line on the half-hour, but managed to hoof the ball to safety.
Irish failed to capitalise from the subsequent lineout and then Leeds powered their way into the home 22 and earned another penalty for offside, which was kicked over by Thomas.
Burrell then scored the first try of the afternoon when he took advantage of Delon Armitage's mistake from a high ball, stepped past Hewat and slammed the ball down by the post.
But just before the break, Hewat took a pass from Jamie Gibson and crossed the Leeds line. Malone added the conversion to leave Irish side 16-13 down and in touching distance.
Leeds controlled the first period of the second half before Irish regained the upper hand.
But with 10 minutes remaining, Goodridge's try knocked the stuffing out of the home team and handed Leeds a famous win.
Leeds Carnegie director of rugby Andy Key told BBC Radio Leeds: "It was an unbelievable test of character and a testament to the players. What you saw today was a performance that was actually cemented by workrate, it was outstanding.
"This team is growing in strength, growing in character, it understands what it needs to do to win games.
"We put in a lot of time and energy into this game because who knew how important it was not only to win but to actually tell ourselves and show ourselves that we've got the performance to compete at this level."
London Irish: P Hewat; J Rudd, E Seveali'i, S Mapusua, D Armitage; C Malone, P Hodgson; A Corbisiero, D Coetzee, F Rautenbach, K Roche, B Casey (c), J Gibson, S Armitage, G Stowers. Replacements: D Paice, D Murphy, J Tideswell, M Garvey, R Thorpe, T Homer, J Lennard, B Stevenson.
Leeds: J Goodridge; H Fa'afili, L Burrell, S Rabeni, L Blackett; C Thomas, S Mathie; M MacDonald, A Titterrell, J Gomez, E Lund, M Wentzel, C Clark, H Fourie, A To'oala. Replacements: P Nilsen, G Hardy, P Swainston, T Denton, R Oakley, A Gomarsall, J Ford, H Paul.
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