Gregg McNally put Leigh in front early on before Atelea Vea crossed over soon after to extend their advantage.
The hosts went in at the break with an 18-0 lead after Samisoni Langi touched down just before half-time.
Josh Drinkwater scored a drop-goal in the second half before McNally raced the length of the field to score again.
Leigh's victory means Widnes Vikings finish bottom of the Super League at the end of the regular season.
A win for Salford would have ensured third place, which would have been their highest league placing for 40 years.
Instead they slipped to a fifth defeat in their final six games before the Super 8s/Qualifiers split and dropped a place to fourth by virtue of Hull FC beating Huddersfield on Friday.
Leigh coach Neil Jukes said:
"We ticked nearly every box tonight. It was a derby and we didn't want to finish bottom on our first season back in Super League.
"It was important for us to get some confidence and build momentum as we go into the next competition (Super 8 qualifiers). I was really pleased.
"Our game management was good as was our defensive effort plus we kicked the ball at the right time.
"We only gave away six penalties in the entire game so when you do that you have a lot of energy. The players kept turning up for each other and defended repeat sets on our line. It has given us something to build on."
Salford coach Ian Watson:
"People will lose their shirt on that performance today. It was disappointing. The first half we were way below the standards we have set ourselves physically.
"Leigh were dominant - their pack was on the front foot and their half-backs were able to control the game on the back of that."
"Our performances have dipped and there's been a drop in intensity since we secured a top-eight finish.
"What we need to do now is realise what we were doing before and rediscover the hunger and attitude we showed earlier in the season which was winning us games.
"For Leigh to score 18 points in that first half in those conditions put them on the front foot."
Leigh: McNally, Dawson, Crooks, Langi, Brown, Mortimer, Drinkwater, Hock, Higham, Maria, Paterson, Vea, Burr.
Replacements: Pelissier, Tickle, Richards, Hansen.
Salford: Evalds, Johnson, Welham, Sa'u, Bibby, Carney, Dobson, Griffin, Tomkins, Murray, Murdoch-Masila, Lannon, Hauraki.
Replacements: Jones, Lui, Krasniqi, Tasi.
Referee: J Smith.
Leigh cruised to a comfortable victory against Salford to lift themselves off the bottom of Super League in their last game before the Qualifiers.