Cory Allen scored the crucial bonus-point try as Ospreys kept up their pursuit of a place in the Guinness PRO14 Final Series.
The centre touched down in the second half to complement earlier scores from Dan Evans, Olly Cracknell and Justin Tipuric.
It leaves Allen Clarke’s side sitting in fifth but they have reduce the gap to Connacht Rugby and Cardiff Blues above them in Conference A as they battle for the final qualifying spot.
Both sides showed strong carrying in the opening exchanges but struggled to protect their own ball – Ospreys’ Nicky Smith and Cheetahs’ Benhard Janse van Rensburg securing early penalties for holding on.
A scoreless opening quarter saw the Welsh outfit kicking long as they looked to control territory in the game.
Ospreys threatened the line with a deep lineout half way through the opening period but their rolling maul was halted by the Cheetahs.
The visitors opened the scoring not long after making the most of Guinness PRO14 top try-scorer Rabs Maxwane dropping Aled Davies’ box kick.
After the ball went through the phases Evans spotted a gap thanks to the overlap, giving a quick dummy before racing in.
It was not long after Luke Price’s conversion when Ospreys doubled their advantage through blindside flanker Cracknell.
After a powerful break from Hanno Dirksen the ball was expertly handled by Davies, Price and lock Adam Beard before it went wide for the back-rower to run in.
The hosts threatened late in the half with a deep lineout but Ospreys won the feed at the scrum after halting their drive for the line.
Cheetahs stormed into the second period breaking from inside their 22 through Van Rensberg after the kick-off.
Justin Tipuric stopped the initial break but the ball was recycled and Dries Swanepoel powered over to give the hosts the perfect start to the half.
But it was undone almost instantly as Tipuric won a penalty from the restart and showed his experience to tap and go, evading the defenders and diving over to make it 21-7 after replacement Sam Davies’ conversion.
Ospreys looked to have secured the bonus point after 52 minutes when Cory Allen dived onto Owen Watkin’s stabbed grubber kick but he was ruled to be in front of the kicker.
It was just four minutes later when he did dive over, making the most of Evans’ offload close to the line to touch down.
Ospreys continued to push for extra points and added another three to their tally 12 minutes before the end through replacement fly-half Davies.
Cheetahs reduced arrears thanks to a late try from Reinach Venter, after a simple catch and drive from a deep lineout to dab down off the back of a maul to make it 31-14.
The visitors almost completed their win in style but No.8 James King, on his 150th Guinness PRO14 appearance, fumbled with the line begging after Sam Cross’ powerful break.
Cheetahs – Tries: Dries Swanepoel, Reinach Venter. Conversions: Tian Schoeman (2).
Ospreys – Tries: Dan Evans, Olly Cracknell, Justin Tipuric, Cory Allen. Conversions: Luke Price (3), Sam Davies. Penalty: Davies.
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