Ethan Pinnock, Keane Lewis-Potter, and Kevin Schade also netted for the visitors, who were full value for the win as Luton showed little fight and quality until a late Luke Berry consolation in a display that will worry manager Rob Edwards.
Brentford move up to 14th with 35 points from 34 matches and, while not mathematically safe, will feel it is almost job done. They are 10 points clear of third-bottom Luton, who have 25.
The visitors might have won by a greater margin but home goalkeeper Thomas Kaminski made several superb saves and the excellent Bryan Mbeumo struck the post as he ran the show for Brentford, who were without injured England forward Ivan Toney.
“We controlled the game from the first minute to the end. We had to win. We know how tight it is to stay in the Premier League,” Wissa said.
“I wanted to score a hattrick, it was the first time I’ve had a chance to score one. It’s unlucky for me. I think Brentford are back, even if it is late (in the season).”
Luton have now conceded 10 goals in their last two matches and a once promising fight against the drop is starting to flounder.
Opening goal
Kaminski produced a superb save to keep out Lewis-Potter’s header but Brentford got the opening goal their early pressure deserved and it was a superb finish from Wissa.
He latched onto a perfectly-weighted through ball from Mbeumo and thundered home a shot from 25m.
Mbeumo struck the bar but Brentford did get a second with another flowing move when his low cross was not cleared and Wissa bundled the ball in from close range.
The third goal arrived just past the hour and it was all too easy as Sergio Reguilon’s corner was headed home by Pinnock before Luton switched off for Brentford’s fourth.
Mbeumo was played into acres of space in the box and his cross to the far post was headed in by Lewis-Potter.
Schade swept home the fifth before the hosts did get a late goal when Berry profited from a poor pass.
“Very low. It’s a tough loss to take,” Luton midfielder Jordan Clark said. “We built this game up so big all week. We were second-best all over the pitch.”
“Full credit to Brentford, we weren’t good enough as a team. I want to say sorry to the fans who were amazing.”
“You have to go home and look in the mirror as an individual player. We’ve got to be men and bounce back next week. We’ve got four massive, massive games now.”
Luton visit Wolverhampton Wanderers next Saturday before facing Everton, West Ham United, and Fulham in their remaining matches this season as they battle to escape relegation.