Watkins's strike in the fourth minute handed the Bees a vital win at Griffin Park with his 25th goal of the season, firing past Declan Rudd.
The victory equals a club record for Brentford of eight successive league wins. The west London club are now just a single point (81) behind second-placed West Brom (82) with two games left to play.
The result leaves Preston in ninth and five points behind Cardiff who occupy the final play-off spot.
Stoke moved a step closer to safety after securing a 1-1 draw against Bristol City which damaged the hosts' play-off prospects. Danny Batth converted a second-half header which cancelled out Filip Benkovic's strike just before the break.
Stoke are now five points clear of the relegation zone with two games left to play, while Bristol - currently managed by caretaker boss Dean Holden - like Preston trail Cardiff by five points in tenth.
Birmingham City - who are currently managerless - moved closer to Championship safety with a last-gasp injury-time equaliser against relegation rivals Charlton Athletic.
Lukas Jutkiewicz teed up Borussia Dortmund-bound youngster Jude Bellingham to slide home a dramatic late goal to cancel out Macauley Bonne's tap-in just before the hour mark (58').
Nottingham Forest are on the verge of securing a play-off place after a 2-2 draw with Swansea. Two Sammy Ameobi goals from outside the box (20' & 55') proved crucial.
Liverpool loanee Rhian Brewster gave Swansea the lead in the eighth minute before Ameobi's first strike, and was then followed up by an Andre Ayew penalty on the stroke of half-time after Ryan Yates was adjudged to have caught the Ghana international.
But Ameobi would have the last laugh as he drilled his 25-yard strike into the bottom corner. Forest are now five points clear of seventh-placed Millwall with two games left to play. Swansea, meanwhile, are three points behind Cardiff in eighth.
Watch the EFL highlights on Quest every Saturday at 9pm, and Wednesdays at 10.30pm for mid-week fixtures. Stream live and catch-up on dplay