The Spaniard, whose rise up the rankings after injury has been one of the more remarkable stories of the season, has never been beyond the third round at Wimbledon in 11 previous appearances.
But he showed he is in great form by defeating grass-court specialist Mahut 7-6 (7/3) 6-1 7-6 (7/5), although the Frenchman did appear to be struggling with injury.
Robredo, 31, has played Murray four times before, with the head-to-head tied at 2-2, but the Spaniard's last win came in 2007 and they have never met on grass.
He said of Wednesday's win: "Brilliant. I think I played well. I played against a guy who is very difficult to play on grass, and I beat him in three sets. I just I have to congratulate myself because today [Wednesday] I did a great job.
"So I'm going to play Andy in two days. Now I prefer to enjoy it, to relax, to go have a great dinner because I deserve it, with a chocolate cake with ice cream. Then tomorrow [Thursday] is going to be a long day to think about Andy."