The Saddlers looked more likely to take the lead in the first half, with Albion goalkeeper Jon McLaughlin keeping out Paul Downing and Romaine Sawyers.
Burton came close when Stuart Beavon tricked his way into Walsall's box but fired into the side-netting.
Wigan's win over Bury means Albion's lead at the top is cut to two points, with Walsall staying in fourth.
Burton Albion manager Nigel Clough told BBC Radio Derby:
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"There was nothing in it at all. It was never going to be an open, free-flowing classic I don't think.
"Chances were few and far between but we got a clean sheet and another point on the board.
"We played better on Monday night, in terms of knocking the ball about, without doubt. But we got back to what the teams been all about for the last season or two - being solid and not giving anything away."
Walsall head coach Sean O'Driscoll told BBC WM 95.6:
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"I think we were organised and disciplined and professional. I've watched a number of games here where teams have fallen into a trap trying to play where there's no space.
"They play in a defensive shape and do it really well, so I thought we created the better chances and it was a different kind of performance to the ones we've put in before.
"Hopefully the chances we created today will go in in the following games."
Burton's lead at the top of League One was cut to two points after they were held by fellow high-flyers Walsall.