The Lions opened up a 20-0 lead inside as many minutes through tries from Stephen Ferris and Keith Earls, with James Hook converting both efforts and kicking two well-taken penalties.
Things were less straightforward for the remaining hour, however, as the Cheetahs hit back with two tries of their own to put the Lions under severe pressure.
The home side nearly sneaked a seemingly improbable victory with just 90 seconds remaining but fate saw Louis Strydom's long-distance drop goal slide narrowly wide of the far post, much to the relief of skipper Paul O'Connell and his men.
Having taken an eighth-minute lead courtesy of a 45-metre penalty from Hook, the Lions were celebrating their first try of the match just two minutes later.
With Jacques-Louis Potgieter having missed a penalty attempt of his own to level the scores, Hook's long drop out was fielded by the Cheetahs but they failed to protect the ball at the ensuing ruck just outside their own 22.
When the ball came loose, Ferris stooped to pick up possession before displaying a fantastic turn of pace to sprint clear. The Ulster flanker scored a 50-metre interception try on Wednesday night but the nature of this effort was arguably even more impressive.
Hook added the simple extras from directly underneath the posts to increase the lead to 10 points as the Lions continued where they left off against the Golden Lions in Johannesburg four days ago.
A good start got even better with just over a quarter of an hour on the clock when Earls claimed the Lions' second try in quick succession.
The Munster centre put to bed a disappointing opening first half against the Royal XV in his first Lions appearance a week ago as he supplied an accomplished finish to a beautifully executed move from the Lions.
Smooth lineout possession on halfway allowed Harry Ellis to provide Hook with quick ball and the Ospreys No10 lofted a deft chip over the head of the onrushing Cheetahs midfield and straight into the arms of Earls.
The 21-year-old stepped past both full back and right wing in the space of 10 metres before diving over the line a similar distance from the left of the posts.
Hook duly converted to give the tourists a 17-0 lead with the same number of minutes played.
The Welsh playmaker, who was only called up to the squad on the eve of departure for South Africa after injury kept Leigh Halfpenny at home for the early part of the tour, then kicked a second penalty to extend the advantage and further silence the home section of the 23,000-strong crowd.
Former Springbok squad member Meyer Bosman was penalised for holding on after Luke Fitzgerald and Halfpenny made had driven the Lions forward and Hook had little difficultly in making the Cheetahs centre pay for his error.
The one-way traffic of the opening quarter soon came to an abrupt end with the game's major turning point arriving with 23 minutes gone.
Try-scorer Ferris made a crunching tackle as the Cheetahs hurled themselves into the Lions 22 but the Irishman failed to roll away and considerably slowed down South African possession in the process.
Guinness Premiership referee Wayne Barnes was not impressed and he responded by sending the Lions' blindside to the sin bin.
By the time Ferris returned 10 minutes later, the Lions' lead had been cut to just six points.
Left wing Danwel Demas was the first Cheetah to cross the Lions' line four minutes after Ferris' untimely departure.
The South African Sevens star looked dangerous throughout the afternoon and he gratefully accepted Potgieter's well-weighted pass just five metres from the Lions' line as the Cheetahs made immediate use of their numerical advantage.
Demas went over untouched, taking the scoring pass just inside the left touchline and then cruising round towards the posts for a score that brought life back into a previously startled South African support.
The 27-year-old flyer then went close to claiming another try of his own with 31 minutes played and, although the bounce of ball proved unkind to his own quest, the Cheetahs didn't have to wait long for their second team score.
This time it was power rather than pace that did the damage, with prop Wian du Preez barging over from close-range after the Lions had lost possession deep inside their own territory.
Fly-half Potgieter was successful with both conversions to bring the Cheetahs back to within a single score and dispel any thoughts of a Lions try-scoring fest to match the 74-10 midweek win at Coca Cola Park.
Hook and Potgieter traded penalties either side of half-time before impressive work in the scrum from the Lions pack earned another kickable penalty attempt.
Hook rewarded his forwards for their dominance in the setpiece with his sixth successful kick at goal from six attempts as the Lions moved nine points clear with just under half an hour remaining.
That proved to be the Lions' last scoring contribution of the match, however, as Britain and Ireland's elite failed to hit their Wednesday night heights.
Instead, it was the Cheetahs who claimed their third try of the match through a 90-metre interception from centre Corne Uys.
With the Lions pressing for what surely would have been the killer score, Shane Williams attempted to feed Hook just 10 metres from the Cheetahs line but Uys read the pass perfectly and ran the almost the length of the field to give his side real hope of a remarkable victory.
With the conversion added from underneath the uprights the Cheetahs knew another penalty or drop goal would see them turn a two-point defeat into a single-point win over McGeechan's men.
And they so nearly ensured that was the case with more than 78 minutes played as Strydom made contact with a 45-metre drop from wide on the left. Both sets of players seemed to have to wait an eternity for the ball to fall from the South African sky and when it did, it was the Lions who were breathing a sigh of relief.
Having seen out the closing minute-and-a-half in Bloemfontein, the Lions now travel to Durban to face the Sharks at ABSA Stadium.
It's then on to Cape Town to play Western Province next Saturday before a Tuesday night clash with the Southern Kings precedes a return to Durban for the first Test against the Springboks on June 20.
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