Earlier, Bangladesh restricted South Africa to 113 for 6 with Taskin's 2 for 19 and support from Tanjim after Proteas skipper Aiden Markram won the toss and opted to bat first.
South Africa were struggling at 23/4 at one stage but Klaasen and David Miller, who was celebrating his 35th birthday, put on 79 runs for the fifth wicket to rescue their side.
Klaasen scored 46 runs off 44 balls with the help of two fours and three sixes before he was clean bowled by Taskin in the 18th over on the score of 102.
Miller, who had hit a match-winning unbeaten 59 in a thrilling win over the Netherlands, was bowled by Rishad Hussain four runs later.
He played an inning of 29 runs in 38 balls with the help of one four and one six.
On Saturday, at the same ground, the Proteas team had scored 12-4 against the Dutch team.