They were made to fight all the way but found the firepower when they needed it to put the South African visitors away with something to spare.
The result moves them 36 points clear of Cardiff Blues, the team in fourth place in the conference, with only 35 points available to the Welsh even if they win every remaining game with a bonus point.
The heavy rain that arrived just before kick off threatened to sour a night to remember for Glasgow Warriors – Rob Harley equalling their appearance record with 177 games while Scotland international squad members Tim Swinson and Scott Cummings returning to the side after injuries – but the players looked determined not to let it deter them from their usual ambitious style.
It backfired spectacularly early on when they lost the ball in attack and Malcolm Jaer, the wing, was put clear after stolen ball and raced clear from his own half, beating Nico Matawalu as the Glasgow wing chased him all the way to the line.
The South African lead did not last long as Glasgow went through the phases and a short pass from Adam Hastings to Matawalu, cutting a clever angle across the defence, saw him cut though on his own for the levelling try.
It took Glasgow another 10 minutes to get their noses in front, thanks to a glorious piece of individual skill from Ruaridh Jackson, the full back. Getting the ball in his own half after Swinson had knocked it clear in a tackle, he chipped over the first line of defence, controlled the ball with his left knee and took the kind bounce to race in for the second try.
Add an exchange of penalties later in the half and the Scots had the advantage at the break, soon cut when Neil Marias, the South African fly half put over his second penalty from only a couple of feet inside the Glasgow half.
Glasgow were still getting the better of the game in open play and a clever chip from Hastings with Matawalu and George Horne, the scrum half, both chipping it further to force Cheetahs scrum-half Shaun Venters, to concede a scrum five.
Glasgow battered away at the line, getting nowhere but won a penalty, kicked for a line out and went for the maul with prop Siua Halanukonuka driving under a pile of bodies for the try that restored the Scots advantage.
The big question was whether they could add the bonus point – they answered it quickly. This time, it was more of a team try with Hastings again chipping into space with Matawalu and Horne forcing the visitors to carry over their own line. Though held out, they won a penalty, kicked for touch and Siua Halanukonuka, the prop, was driven over from the maul.
It secured the bonus point but was a long way from securing the result with the Heineken Cheetahs taking the game back to the Scots and taking the game in to the Scots 22 where a huge cut-out pass from Shaun Venter put Clayton Blommetjies, the full back, over for a score that brought the visitors back to within four points.
Again, the Glasgow ability to strike from long range earned them a the cushion they needed, Nick Grigg, the centre, cutting through on his own from his 22 to make the Cheetahs half on his own, and finding Henry Pyrgos in support for the fifth try.
Glasgow Warriors - Tries: Matawalu, Jackson, Halanukonuka, Horne, Pyrgos. Con: Hastings (3). Pens: Hastings (2)
Cheetahs - Try: Jaer, Blommetjies. Cons: Marias (2). Pens: Marais (3)
Glasgow Warriors: R Jackson; L Jones, N Grigg, S Johnson, N Matawalu; A Hastings, G Horne; A Allan, J Malcolm, S Halanukonuka, T Swinson, S Cummings, R Harley, C Fusaro, M Fagerson.
G Stewart for Malcolm, 73
O Kebble for Allan, 58
D Rae for Halanukonuka, 60
M Smith for Fusaro, 58
A Ashe for Swinson, 69
H Pyrgos for Horne, 60
B Thomson for Jackson, 73
R Tagive for Jones, 41
Cheetahs: C Blommetjies; C Barry, F Venter (C), N Lee, M Jaer; N Marais, S Venter; C Marais, T Van Jaarsveld, L de Bruin, J Basson, R Hugo, H Venter, O Mohoje, U Cassiem.
Sin bin: F Venter, 79-end
J Du Toit for Van Jaarsveld, 65
O Nche for Marias, 65
J Coetzee for de Bruin, 31
C Wegner for Basson, 58
P Schoeman for Mohoje, 47
Z Mkhabela for S Venter, 76
F Zeilinga for N Marias, 69
Referee: G Clancy