Leicester led at half-time but Toulon turned up the pressure in the second-half, much to the delight of the Stade Felix Mayol crowd.
And Bastareaud admits the home support drove his to victory.
"The city had been waiting for this game since we qualified for the quarter-finals. But I have never experienced a feeling like that.
"This club and city is enormous. We gave everything today for a public emergency. This is history.
"What changed after the break? We simply made fewer mistakes.
"We knew that the referee was very picky and we did not panic either. It was a tough battle against a very strong team."
Scrum-half Sébastien Tillous-Bordes echoed Bastareaud's sentiments and added: "It puts chills. It was important to have the public behind us in a tough match. The 16th man did the job. '
Toulon temporarily return to Top14 action before turning their attention back to the Heineken Cup.
They face Saracens at Twickenham on April 28 and the France powerhouse admits it will be a huge encounter.
"We must look to Twickenham now but first we will enjoy this win. Then we will prepare for the match against Saracens. It's magical, huge!"