Thurston reached base in four of six plate appearances, singling in the first inning, tripling home two runs in the second and slugging a three-run homer in the third.
The 27-year-old second baseman said he realized as he came to bat in the fifth that he needed only a double to hit for the cycle.
"I was aware of it, I'm not going to lie," said Thurston, who walked before grounding out in the seventh and ninth. "But they made some pretty good pitches away the rest of the night, where I wasn't able to get it."
The former fourth-round draft pick, who also drove in five runs on April 21 against the Yankees, said his big game was the result of sticking with his regular approach at the plate.
"I've been seeing the ball pretty well lately and I was just trying to come through in situations where we needed it, with runners on base, and I was able to do that tonight," he explained.
Chris Roberson had three hits, scored three runs and plated two, while Pedro Swann and Jesus Merchan each chipped in two hits and two runs scored for the Lynx (35-51).
Reliever Geoff Geary (1-1) picked up the win, yielding one run on three hits while striking out two in two innings. Fabio Castro notched his first save after allowing one hit in three scoreless frames.
Scranton/Wilkes-Barre starter Steven White (3-2) lasted 2 1/3 innings and surrendered eight runs -- seven earned -- on nine hits with a walk and a strikeout. Ben Kozlowski surrendered one hit in four scoreless frames.
Bronson Sardinha and Chris Basak ripped back-to-back homers in the fourth for the Yankees(46-41). Sardinha hit a towering three-run blast, his eighth, past the right-field foul pole and Basak followed with his sixth of the season.