The Warriors played some excellent rugby in the first-half and deservedly went 20-10 ahead at the break through Adam Ashe and George Horne tries.
Adam Hastings also added 10 points from the tee while the hosts’ response came via Toby Arnold’s score and Jonathan Pelissie’s penalty.
Although Noa Nakaitaci’s effort gave Lyon hope in the second period, Stuart Hogg, Huw Jones and Ashe touched down to secure an important win for the Scottish outfit.
In their debut in the competition, Lyon have yet to garner a win, despite being in decent form in the Top 14. They have impressed domestically over the past couple of years, but Cardiff Blues, Saracens and now Glasgow have shown the Frenchmen that a European campaign is an altogether different proposition.
They have been outplayed by all those sides and have not learned the lessons from the defeats. In the opening rounds, Pierre Mignoni’s men failed to deal with the tempo of the PRO14 and Premiership teams and they once again failed to adapt in Round Three.
Horne and Hastings enjoyed plenty of quick ball at half-back and, as a result, Tommy Seymour was a constant threat in the early stages. Although they were unable to cross the whitewash, their fly-half rewarded their dominance with a three-pointer.
Lyon did threaten soon after when Arnold’s break had the away team backtracking, leading to Pelissie levelling matters from the tee.
Dave Rennie’s charges were controlling matters, however, and Hastings restored their advantage via a penalty before they deservedly touched down for the first time in the match through Ashe, who benefited from George Horne’s good work.
To the hosts’ credit, they hit back and reduced the arrears as Arnold scampered over unopposed in the right-hand corner.
That score kept them in the contest but they were soon back under pressure when Nick Grigg broke upfield and almost created a try, but Pete Horne’s poor pass halted the momentum.
Unperturbed, they remained on the front foot and secured a 10-point buffer at the interval after scrum-half Horne sniped across the line.
Glasgow maintained the momentum in the second period and deservedly increased their lead through the boot of their pivot.
Mignoni’s men did respond when Nakaitaci crossed the whitewash after good work from Lionel Beauxis, but Hogg secured the win after Charl McLeod failed to deal with Pete Horne’s grubber through.
McLeod partially made up for that error by touching down, but not before Jones had given the Scottish outfit the full five points by scoring under the posts.
The Top 14 side took risks late on to try and rescue something from the game and their over-ambition led to Ashe going over for the Warriors’ fifth try.
Tries: Arnold, Nakaitaci, McLeod
Cons: Pelissie, Doussain
Tries: Ashe 2, Horne, Hogg, Jones
Cons: Hastings 4
Pens: Hastings 3
Yellow Card: Swinson
Lyon: 15 Alexis Palisson, 14 Toby Arnold, 13 Rudi Wulf, 12 Adrien Seguret, 11 Noa Nakaitaci, 10 Lionel Beauxis, 9 Jonathan Pélissié, 8 Carl Fearns, 7 Liam Gill, 6 Julien Puricelli (c), 5 Manuel Carizza, 4 Felix Lambey, 3 Hamza Kaabèche, 2 Mickael Ivaldi, 1 Albertus Buckle
Replacements: 16 Virgile Lacombe, 17 Alexandre Menini, 18 Kévin Yaméogo, 19 Hendrik Roodt, 20 Patrick Sobela, 21 Charl McLeod, 22 Pierre-Louis Barassi, 23 Jean-Marc Doussain
Glasgow: 15 Stuart Hogg, 14 Tommy Seymour, 13 Nick Grigg, 12 Peter Horne, 11 DTH van der Merwe, 10 Adam Hastings, 9 George Horne, 8 Matt Fagerson, 7 Callum Gibbins (c), 6 Adam Ashe, 5 Jonny Gray, 4 Rob Harley, 3 D’Arcy Rae, 2 Fraser Brown, 1 Oli Kebble
Replacements: 16 George Turner, 17 Jamie Bhatti, 18 Siosiua Halanukonuka, 19 Tim Swinson, 20 Chris Fusaro, 21 Ali Price, 22 Huw Jones, 23 Niko Matawalu
Referee: Matthew Carley (England)
Assistant referees: Greg Macdonald (England), Wayne Falla (England)
TMO: Geoff Warren (England)