They now need one point from their final group game - against Howard's former club Clermont Auvergne in France next Friday night - to march on.
Tries during the closing seven minutes from lock Louis Deacon and centre Dan Hipkiss, both converted by fly-half Andy Goode, who finished with 19 points, left Stade distraught.
"We are in the entertainment business, aren't we?" said an elated Howard, following Leicester's thrilling recovery from 22-12 adrift.
"There is an amazing amount of belief at the club, and the players showed great conviction. Stade are a quality team, one of the sides who are capable of winning the Heineken Cup.
"At 22-12 behind, I didn't think the game was gone. We knew that 10 points was in our reach, and the players really believed in themselves."
Newcastle Falcons director of rugby Rob Andrew had some harsh words for his side despite a 47-20 win over Border Reivers.
After a disappointing first-half performance which saw Newcastle go in at half time a point behind, the Falcons got their act together in the second half to score 28 unanswered points.
"I'm a bit disappointed because every time we scored in the first half we switched off and let them score," said Andrew.
"We got our five points which is important but with all due respect I thought the Borders were pretty poor and we could have had a 60 or 70-point victory.
"I thought defensively we were pretty lazy and I think they thought it was going to be too easy."
The victory guarantees Newcastle qualification to the quarter-finals of the competition, and a win over Brive next week would see them go through as one of the top two seeds.
Borders boss Steve Bates, making a return to his former club, was pleased with the spirit shown by his young side.
"We always knew this was going to be a tough game, and I was really proud of our lads for the effort and commitment they showed," said Bates.
"There are plenty of positives to take out of the game."