The Free Staters entered the clash knowing that they needed to win all four of their remaining pool matches, preferably with bonus points, to have a shot at repeating their feat from last season of reaching the top six.
But the Warriors’ effective display was enough to secure them a bonus-point victory and keep them in the top spot on the Conference A log.
The hosts produced a well-balanced performance in the first half and they did well to make a few vital line breaks, which allowed them to apply consistent pressure on the Toyota Cheetahs.
Glasgow showed their intent early on with a try by Callum Gibbins, but the Free Staters’ William Small-Smith brushed off a defender minutes later to score their first try in the 14th minute.
The Glasgow Warriors responded with back-to-back tries by Kyle Steyn and Rory Hughes to stretch their lead to 21-5.
The Toyota Cheetahs were reduced to 14 men shortly before halftime as Charles Marais infringed in a scrum, but they managed to shut out the opposition going into the break.
The visitors earned the first points in the second half as Joseph Dweba crashed over the tryline from a powerful run to trail 21-12.
But they were unable to stop the hard-running home side from earning their bonus point, as Stafford McDowall touched down and then Scott Cummings broke through for their fifth try with five minutes to play.
The Toyota Cheetahs, however, had the final say in the match as Small-Smith scored his second try from a turnover close to the Glasgow Warriors’ tryline, which saw them go down 35-17.
In the other matches, the Cardiff Blues defeated the Scarlets 41-17 at Cardiff Arms Park, Connacht beat Benetton 29-14 in Galway, Edinburgh registered a 28-11 victory against Leinster at BT Murrayfield, Ospreys held their nerve against the Dragons for a 29-20 win in Swansea, and Munster overcame Zebre Rugby Club 31-12 in Limerick.
Glasgow Warriors 35 (19) - Tries: Callum Gibbins, Kyle Steyn, Rory Hughes, Stafford McDowall, Scott Cummings. Conversions: Pete Horne (3), Adam Hastings (2).
Toyota Cheetahs 17 (5) – Tries: William Small-Smith (2), Joseph Dweba. Conversion: Tian Schoeman.