South Africa had battled its way to 207.
Bangladesh looked on course to record an upset win in its first World Cup appearance.
Its opening pair had put on 69 but Khaka intervened.
The 29-year-old right arm medium pacer broke the partnership, picking up three quick wickets to derail the chase.
She returned later to nab a fourth – her 100th ODI wicket.
Khaka finished with four for 32 and Bangladesh was bowled out for 175.
Earlier, South Africa lacked urgency at the top in the absence of regular opener Lizelle Lee.
Tazmin Brits took 19 balls before she got off the mark with a boundary over midwicket.
Her frustration did not last much longer. She lobbed a very gentle catch to Rumana Ahmed for eight.
Laura Goodall replaced her at the crease and renewed the struggle.
The inactivity was confined to one end, though. Laura Wolvaardt scored in a way which was hard to reconcile with the slumber her team-mates endured.
But on 41, she tried to work a straight delivery from Ritu Moni into the legside and was bowled.
Bangladesh got further ahead when Goodall slapped a reverse sweep to Ahmed.
South Africa had slumped to 69 for three and had used up 20 overs.
Just as the side stretched for the acceleration pedal, the captain Sune Luus was run out for 25 in most unfortunate circumstances.
Ahmed got a deflection from a straight drive. She had quite a match.
Three catches – one off her own bowling – and that lucky, well, misfield.
Eventually a more productive partnership stuck. Chloe Tryon (39) and Marizanna Kapp (42) put on 71 for the sixth wicket to edge their side closer to 200.
Bangladesh strike bowlers Jahanara Alam (two for 28) and Fariha Trisna (three for 35) handed over a good position to the batters.
Openers Shamima Sultana and Sharmin Akhter had the comfort of time as they set off in pursuit of history.
They made use of the resource and built slowly.
But Sultana tried to hoick a yorker from Khaka and was bowled for 27.
It was a small opening and South Africa set about driving in a wedge.
And Khaka had a firm grip on the hammer. She nicked out Akhter for 34 and Murshida Khatun fell two balls later for a duck.
A lot of hard work had unravelled in a couple of overs. The pressure had shifted to Bangladesh.
Tryon looped in some very cheap overs of left arm spin and the panic seemed to materialise into something weighty and concrete.
The run rate had crept up to more than five and, in the context of the game, that was a stretch.
Masabata Klaas run out Fargana Hoque with a direct hit and the sky collapsed a little more around the Bangladesh camp.
The Bangers battled on but victory dipped below the horizon when Khaka returned for her third stint and removed Ahmed for 21.
But they did muster a flurry of boundaries at the death to cut the deficit.
