Warriors trailed 17-10 at half-time in South Africa, but Robert Harley's late try snatched victory making it six wins from six in the PRO14.
Speaking to BBC Scotland after the game, Rennie said: "We took a little while to adjust to the altitude and the effort it takes to play at pace. But we showed composure and we hung in there. We are rapt with the impact we got off the bench and how hard the guys worked for each other.
"We needed some guys to empty the tanks and then get fresh legs out there. The Cheetahs try to run teams off their feet at altitude but we are a very fit side and felt we could put them under a lot of pressure.
"We had our battles at line-out time and probably didn't build phases enough in the first half. But we did better at crucial times in the second half and managed to win an arm wrestle in the end."
Rennie brought his Chiefs head of strength and conditioning coach Phil Healey with him to Glasgow and believes the squad are reaping the benefits of some intensive pre-season work.
"The Chiefs have always been a very fit side, and that has been our mindset here [at Glasgow]," added Rennie.
"Phil has worked really hard with our trainers and the guys are in great nick. I think there were 28 PBs with regard to conditioning stuff, so the boys are a lot fitter than they have been in the past, and that was important here.
"We needed the boys to lift to another level, and I think the mind can play tricks on you at altitude. We will learn from that and be far better the next time we come to Bloemfontein.
Attention now turns to Europe, with an opening Champions Cup pool match at English champions Exeter next Saturday.
Glasgow's squad faced a four-hour bus journey to Johannesburg after Saturday's match - "There is a massive music festival in Bloemfontein, and no accommodation," explained the head coach - before their return flight to Glasgow.
"It will be another big test against Exeter next week," he added.
"We won't get home until early on Sunday, so recovery will be crucial for the next couple of days."