Lineen's men trailed 14-3 at half-time after Maxime Medard and Virgil Lacombe crossed for the hosts, but a try from flying Dutchman DTH van der Merwe fired them back within touching distance at 14-10.
But Toulouse took no prisoners as they scored three tries in the last 25 minutes to leapfrog Wasps into top spot in Pool Six and condemn Glasgow to their third straight European loss, leaving them 11 points behind the English side with only a maximum of ten left to play for.
Glasgow had endured a nightmare in the build-up to the game, which was postponed more than 75 hours after delays and missing luggage caused chaos.
After such a disrupted build-up to the game, Lineen was pleased with the way his players responded, saying: "It was a tough game, we were in it for 50 minutes and the boys worked hard but the set-piece wasn't strong enough.
"Playing with 14 men for 20 minutes also had a big effect on the game. A team in control of the set-piece has the momentum.
"But there were some fantastic performances after a challenging five days. It's had a huge impact. We have worked hard, dug deep and made the right decisions but unfortunately we weren't strong enough."
A fired-up Toulouse opened the scoring through Medard after ten minutes before Lacombe gave the hosts a 14-3 half-time lead, Glasgow's points coming from the boot of Ruaridh Jackson.
Van der Merwe gave Glasgow a glimmer of hope before Toulouse put them to the sword with tries from Clement Poitrenaud, Thierry Dusautoir and Vincent Clerc.
But Lineen remains adamant that his side can enter crunch back-to-back Magners League encounters with Edinburgh with confidence despite the loss and added: "Johnny Barclay, Rob Harley, Richie Grey were all great and Max Evans and Graeme Morrison were outstanding. We can take great heart from that."