The 2017 Champions Trophy winners showed signs of resurgence against the Proteas, with openers Fakhar Zaman and Babar Azam adding 81 for the first wicket and fast bowler Mohammad Amir taking his wickets tally to 15 for the tournament.
Mahmood was pleased with Pakistan's all-round display heading into the business end of the tournament but warned his players it will count for nothing if they do no take their chances in the opening powerplays against New Zealand.
"They are a very strong side and have won all their games," Mahmood told a news conference on Tuesday.
"They've got match winners in their side, so it's going to be crucial for us to take wickets with the new ball. And when we bat, we need to get runs up front.
"If we can get our discipline right like last game - in batting, bowling and fielding - then we can beat any side."
Amir's knack of picking up wickets with both the new and old ball has been key to Pakistan's qualification hopes.
Mahmood believes the 27-year-old's form after a difficult start to 2019 could be a reference point for other bowlers in the dressing room, especially fellow paceman Hasan Ali.
Ali has come under heavy criticism after conceding 84 runs in nine overs in a defeat by India at Old Trafford and was subsequently dropped against South Africa.
"I think everyone in cricket, whether you're a batsman or bowler, goes through good and bad form," Mahmood added.
"So he's low on confidence. But he's running in, and working hard on and off the field. It's just a matter of getting wickets, and he'll be back.
"The same thing happened with Amir, you know. After the Champions Trophy, he was struggling - but still bowling well. Now you can see that he's having a good time."
(Reporting by Hardik Vyas in Bengaluru; Editing by Hugh Lawson)
