Crewe seized the lead when Marcus Haber tapped in a rebound in the first significant opening of the game.
Brad Inman then fired at goal from 20 yards, stinging the palms of Jon McLaughlin as Crewe sought a second.
But Lucas Akins grabbed the equaliser for the Brewers when, from Mark Duffy's cross, he turned three Crewe defenders and slotted past Ben Garratt.
Crewe, who lost defender Jon Guthrie late on with a cheekbone injury, remain seven points from safety with 11 games left to play.
The Brewers maintain their four-point advantage over second-placed Wigan, now nine points clear of third-placed Gillingham.
Crewe manager Steve Davis told BBC Radio Stoke:
"Drawing against the team that is top of the league should give us confidence because it shows that on our day we can compete with the very best.
"But, in the context of our situation, we need to be getting three points, especially from our home games. It is still a good point and I don't think it's two points lost.
"Overall a draw was fair. There was not a lot between the two sides. We took the lead again but we didn't hold onto it long enough. We have taken the lead in 17 games and only gone on to win six, so that tells its own story."
Burton boss Nigel Clough told BBC Radio Derby:
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"Jon McLaughlin's not had much to do.  We should have worked their goalkeeper a bit more.
"We had some brilliant situations today which we didn't turn into clear cut chances and ultimately we were just lacking that little bit of quality in the final third to get us the winner.
"I was pleased the way the players responded to going a goal down and the way we got back into the game and deserved out equaliser and were looking to win the game but when you're a goal down away from home then you end up with a point, it's not a bad result."
League One strugglers Crewe held leaders Burton Albion to a draw in a game of few clear-cut chances.