Allen belted a grand slam in the fourth and a two-run double in the fifth. With the game tied at 10 in the ninth, Jason Pridie walked with one out in the A's ninth against Marcos Mateo, reached third on Chris Carter's double and scored on Allen's groundout.
The Cubs opened a 3-0 lead when Darwin Barney hit a two-run double in the second and another run scored in the third on a throwing error by shortstop Cliff Pennington.
Chicago lefty Jeff Beliveau walked the first two batters he faced in the fourth, and Josh Reddick hit an RBI single to pull Oakland within two runs. After walking another batter to load the bases, Beliveau was pulled in favor of Dae-Eun Rhee. Allen roped a 3-2 pitch from Rhee to right field to give the A's a 5-3 lead.
Welington Castillo, competing for the Cubs' backup catcher's spot, hit a leadoff homer in the fifth.
Chicago rallied with a five-run sixth, scoring three on Adrian Cardenas' bases-clearing triple. Cardenas then scored on Jae-Hoon Ha's single to tie the game at 9.
With the game tied at 9 in the eighth, Oakland had Josh Donaldson at third with one out, and he scored when Brandon Moss grounded out to first baseman Anthony Rizzo. But Chicago tied it in the eighth when Dave Sappelt scored on Ha's infield single.
A's starter Brandon McCarthy gave up three runs (two earned) on four hits and one walk over three innings. Chicago's Rodrigo Lopez faced the minimum in two innings.
Up next for Cubs: Ryan Dempster, who was the Opening Day starter in 2011, will make his Cactus League debut on Monday, when the Cubs face the Athletics for the second time in as many games at 2:05 p.m. CT on Monday. Dempster is coming off a 10-14 season in which he totaled 200 innings for the fourth straight year. The game will be broadcast on Gameday Audio on MLB.com and on WGN in Chicago.
Carrie Muskat is a reporter for MLB.com. She writes a blog, Muskat Ramblings, and you can follow her on Twitter@CarrieMuskat This story was not subject to the approval of Major League Baseball or its clubs.