That momentum carried into the fourth set, with Munar being broken in the opening game and two games later Norrie suffered the same fate.
So the match went into a deciding fifth set and Munar was broken in the first game of it, then again in the fifth game.
Daniell and Koolhof held their nerves from there and ended up comfortably getting to the finish line.
In the third round Daniell and Koolhof will play against Mexico's Santiago Gonzalez and Aisam-Ul-Haq Qureshi from Pakistan in the third round.
Given that Daniell lost in the first round at Wimbledon last year, making it to the round of 16, should move him up from his current ranking of 49 to around 41, even if he was to lose his next match.
Meanwhile, Artem Sitak has continued his run in the mixed doubles.
On Saturday he and his partner Laura Siegemund from Germany knocked out the No 13 seeds Rohan Boppana from India and Belarusian Aryna Sabalenka 6-4 6-4, with the finally coming from the fifth match point on Sitak's serve.