The tourists came within a whisker of blowing a 20-point first-half lead as Louis Strydom's last-gasp drop goal effort slid inches wide.
It would have been rough justice on McGeechan's men, who just about deserved their third win of the tour thanks to an explosive opening 20 minutes.
The men in red, fielding their youngest-ever three-quarter line, had to grow up fast in this torrid encounter in the heart of Afrikaaner country.
This was light years away from Wednesday night's walk in the Ellis Park against the toothless Golden Lions. It was what Lions tours are all about as the Super 14 side proved a tenacious and committed opponent.
If the Cheetahs had landed one of the four kicks they missed, it would have been a different story.
Keith Earls and Stephen Ferris put the visitors into a commanding early lead with well-taken tries but the Cheetahs were always going to come roaring back.
Lions head coach McGeechan said: "We knew the games were getting progressively tougher and we expected that.
"We said every player in the squad would get a start and nine of our players were getting their first Lions start.

"You can't under-estimate how important getting a win is for the squad. It became a stop-start game and that was frustrating."
The Lions withstood a ferocious second-half surge and will travel to Durban today with their pride, and winning streak, intact. Few players enhanced their claims for a Test spot yesterday in front of another disappointing crowd.
But the value of recording a third win on the bounce will do wonders for the spirit of this tight-knit bunch of tourists.

England's cricketers may have wilted feebly in the face of an Orange onslaught at Lord's on Friday, but this lot are made of sterner stuff.
Another disappointment was that this bruising contest was witnessed by so few. For a supposedly rugby-mad nation the attendances on this tour have been pathetic.
Maybe all their rugby lovers were stuck in West London pubs watching the action on Sky. More likely they had already endured the dire pre-match entertainment on offer at the Vodacom Stadium and decided to stay away.
Yesterday we were treated to 20 orange-shirted Hell's Angels on Harleys and a Boer singer who would have been booed off stage at Brian Potter's Phoenix Club.
The surprisingly unintimidating atmosphere allowed the Lions to settle into their stride quicker than their opponents.
Ferris latched on to a stray ball at the side of a ruck to gallop home after the flawless James Hook had put them 3-0 in front with a penalty.
Fly-half Hook, who scored 16 points with six from six, then set up a try for centre Earls with a clever chip over the top that the young Irishman raced on to before stepping two defenders.

It was an important moment for the 21-year-old Munster player, who endured a Phokeng nightmare debut in last week's tour opener against the Royal XV.
His midfield partnership with Luke Fitzgerald had the look of a dirt track pairing yesterday but at least young Earls will feel like a Lion now.
At 20-0 in front after 23 minutes, the Lions were rampant but the otherwise impressive Ferris blotted his copybook when he was sin-binned for killing the ball. Danwel Demas and Wian du Preez both crossed as the Lions shipped 14 points with the blindside flanker off the field.
Hook calmed nerves before the break with his second penalty but the second half was all Cheetahs as the Lions struggled badly at the breakdown. Joe Worsley will not play openside again on this tour.
Lions skipper Paul O'Connell said: "We stopped playing a little bit and turning the ball over killed us. Maybe we fell asleep. It was disappointing that we got off to such a good start but let it slip. It's going to take a bit of time and it's time we maybe don't have."
Jacques-Louis Potgieter and Hook exchanged penalties before Corne Uys intercepted Shane Williams' pass and galloped home to make it a two-point ball game.
Lions assistant coach Shaun Edwards said: "We got off to a flying start but then Ferris got sin-binned and they got two tries against us during that period."
Replacement Strydom had a chance to earn a place in Highveld history but his mammoth 45-metre effort with just over a minute left drifted wide.
SCORERS - Cheetahs: Tries: Demas, Du Preez, Uys. Cons: Louis-Potgieter 2, Strydom. Pen: Louis-Potgieter. Lions: Tries: Ferris, Earls. Cons: Hook 2. Pens: Hook 4.
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