Rasmus hit a two-run drive to straightaway center with one out in the second.
With a 1-for-4 day, Colby Rasmus improved to .089 (4 for 45) at new Yankee Stadium.
In the bottom of the inning, Rasmus made a leaping catch to rob Alonso of a homer.
Phelps (1-1) took about 5 minutes to warm up then gave up an RBI single to Rasmus.
Rasmus Elm's 90th minute strike proved a mere consolation for the home side.
Toronto's Colby Rasmus had a two-run homer among his season-high four hits and robbed Yonder Alonso of a solo shot.
Anthopoulos had already begun implementing the plan in 2011, when he pocketed Colby Rasmus in a trade with the Cardinals.
FORBES: GM Anthopoulos Has Blue Jays Back In The Game - Cheaply
The Wolves keeper also had to be alert moments later to tip over Rasmus Elm's deflected shot as the Swedes finished stronger.
Copenhagen is also home to 10 Michelin-starred restaurants and the best young chef, Rasmus Kofoed, gold medallist at the prestigious Bocuse d'Or.
Louis had drafted Rasmus in the first round in 2005 but ultimately decided that his attitude was undermining his talent, which they loved.
FORBES: GM Anthopoulos Has Blue Jays Back In The Game - Cheaply
Phelps didn't allow another hit after Rasmus until Encarnacion hit his second long ball of the night leading off the sixth, a drive off the back wall of the Toronto bullpen.
Colby Rasmus threw home when Wells bluffed a start at third, but Rasmus' one-hop throw bounced off Arencibia and into the Yankees' dugout for an error, allowing both runners to score.
Colby Rasmus hit a leadoff double in the second and later scored on Kawasaki's sacrifice fly, but the Yankees cut the gap in half on Stewart's leadoff homer in the third, his first.
Rasmus scored Lind with a broken bat single to center and David Phelps replaced Logan, but gave up a two-run double to Brett Lawrie, whose liner down the third base line stayed fair by inches.
But Cody Rasmus hit a two-run home run off Rays reliever Jake McGee in the seventh inning, and Toronto tied it in the eighth when Melky Cabrera and Jose Bautista hit back-to-back doubles off Kyle Farnsworth.
After Adam Lind had the first of his two doubles, Colby Rasmus hit a sharp liner that Curtis Granderson, playing right field for the first time in his big league career, got spun around on and Lind scored.
Yankees starter Hiroki Kuroda, who beat Baltimore with a five-hit shutout in his previous start, allowed just two hits through the first seven innings, but was replaced by David Robertson after Colby Rasmus' one-out single under Overbay's glove in the eighth.
Julie and Rasmus seem more interested right now in building up their musical skills and practising their vocals but just before I leave the youth club, to give my ears a break from the screeching electric guitars, Julie says she would like to add something to the comments she has given me.
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