Graeme Jones' men were up against it from as early as the fourth minute when Marley Watkins bundled home from close range.
City were worthy winners against a disappointing Hatters side, with Lee Johnson's men even having the luxury of being able to miss a first-half penalty.
The loss saw Luton drop to second bottom in the Sky Bet Championship, with Barnsley edging above them thanks to a 0-0 draw at Swansea City.
Here's how the Luton players rated in the Ashton Gate reverse.
9. Kazenga LuaLua 4
He gave away the penalty while playing as wing back, which didnt suit him. He tried to commit defenders in the second half but didnt have the same joy as against Fulham | freelance Photo: Liam Smith
10. James Bree 3.5
The left back was a soft touch for the opening goal as he let Marley Watkins bully him out of the way to head in | freelance Photo: Liam Smith
11. Martin Cranie 5
Made his first appearance since getting injured on November 9, replacing Luke Bolton at half time, to help stop the threat from Eliasson. His experience and nous will be needed if Town are to plug holes in their leaky defence | freelance Photo: Liam Smith
12. George Moncur 4
Came on for Tunnicliffe in an attempt to give Town attacking impetus, but six minutes later Bristol scored their third. Any hope of a fightback was extinguished, and he didn't affect the game from there | freelance Photo: Liam Smith