Lawrie followed against David Phelps by lacing a pitch over the third-base bag and down the left-field line, scoring the go-ahead runs.
He later came home with a head-first slide on Melky Cabrera's hit to close out the four-run frame.
Cabrera finished 3-for-4 while Arencibia also smacked a solo homer as Toronto avoided a three-game home sweep.
Brett Cecil (1-0) earned the victory with 1 1/3 innings of scoreless relief.
Chris Stewart went 2-for-3 with a solo homer for New York.