McGregor needed to make a fine flying save to tip away a sharp shot on the turn from Brazilian-born forward Cacau.
Cacau fizzed an angled shot narrowly wide before Rangers began to assert themselves in the final 15 minutes of the first half.
In the day's other German Bundesliga match, Germany striker Cacau scored the only goal as Stuttgart claimed a narrow 1-0 win over Nuremberg.
Stuttgart had warmed up for the match by hitting five goals past Cologne at the weekend, with striker Cacau helping himself to four of them.
The home side took an 18th minute lead when Cacau's cut-back found Pavel Pogrebniak, who dispatched a powerful right foot shot past goalkeeper Allan McGregor.
Blackburn boss Sam Allardyce wants to tie up a deal for Portsmouth striker Aruna Dindane this week and then follow it up with a move in the summer for Stuttgart forward Cacau.
But the Fulham keeper could do little about Germany's third, Mueller tucking in off the post from the edge of the area, and even less about their fourth, substitute Cacau tapping in from Ozil's centre.
Soon after, McGregor kept out a stinging effort from Cacau and did brilliantly to stick out a leg and divert Pogrebniak's scuffed follow-up, with the big Russian striker allowed to carry on from a clearly offside position.
The home side had chances to double their lead before the break, with only goalkeeper Victor Valdes depriving Cacau of a second -- although Lionel Messi did strike the post with a long-range effort for the visitors.
Lukas Podolski and Miroslav Klose continued their remarkable scoring record at international level with a goal apiece to take their joint tally to 88, Thomas Mueller and Cacau added the gloss, while in playmaker Mesut Ozil they appear to have found a real superstar.
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