The visitors nearly edged ahead after nine minutes when Jimmy Phillips cut inside from the left and unleashed a testing low shot which Pompey goalkeeper Trevor Carson gathered at the second attempt.
Dominic Knowles should have done far better than the tame header he managed from Zeli Ismail's wicked cross from the right in the 21st minute.
And Ismail then found Billy Kee in space 15 yards from goal, only for him to sky Burton's best chance 10 minutes later.
Patrick Agyemang should have opened the scoring for Pompey in the 51st minute when he wriggled free in the Burton box, but dragged his effort against the foot of the post.
Sonny Bradley's diving header from the resulting corner, taken by Nicky Shorey, was wide.
Adam McGurk netted for Burton after latching on to Phil Edwards' searching ball into the box, but his strike was ruled out for offside as both teams suffered a night of frustration in front of goal.