Yanic Wildschut put the Latics in front but the U's levelled shortly after through Alex Gilbey's 25-yard strike.
Wigan swiftly restored their lead with Ryan Colclough's calm finish, but Colchester negated the winger's effort with Elliot Lee's curling shot.
George Moncur converted a penalty after he was fouled to make it 3-2, but Grigg's injury-time goal earned a draw.
Wigan manager Gary Caldwell told BBC Radio Manchester:
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"That wasn't good enough. When you squander as many chances as we did then you have to be aware that the opposition are going to come back into the game.
"Second half the performance was nowhere near what we expect. I warned the players that you show your superiority on the scoreboard.
"We'll have to work hard to put it right next week."
Will Grigg's late strike secured a point for Wigan at bottom club Colchester in a thrilling encounter.