Ash nipped in to beat both Bradley Watkins and Michael Toner to a looping ball into the area to head the home side in front midway through the first half.
Michael Cheek missed a golden chance early in the second period, one the Clarets would rue he missed Grubb arrowed the ball into the bottom corner to double Weston's advantage, before Ash tapped in from close range to all but wrap up the points.
Lee Sawyer scored from the penalty spot two minutes into injury time after Mark Hughes was brought down in the area, but it was scant consolation for the Clarets who failed to reclaim a play-off spot.
City came into this game in the West-Country off the back of three straight wins which had momentarily seen the Clarets climb as high as third in the league after the 6-2 win over Farnborough last weekend.
Their hosts on the other hand were hovering just three points above the relegation zone, but had seen an upturn in their recent fortunes, winning three out of four games over the Christmas and New Year period.
After Chelmsford’s recent run of good form it was no surprise to see Mark Hawkes name an unchanged side, sticking with the three centre backs of Michael Toner, Marvin Ekpiteta and on-loan MK Dons man, Harry Hickford.
Following a quiet start to proceedings City were indebted to another loanee, Bradley Watkins, for preventing the home side from taking an early lead. Jamie Edge jinked past the challenge of two Clarets defenders to pull the ball back for Dayle Grubb whose low effort was superbly turned wide by Watkins.
Watkins though may have rued his decision making which lead to Weston taking the lead midway through the first half after choosing to come for a ball he was unlikely to get.
After some good link-up-play by the hosts, between Oliver Watkins at Ollie Barnes, the former cut inside to try and curl a shot towards the top corner which look a wicked deflection.
The deflection caused the ball to loop up into the danger area, tempting Bradley Watkins off his line to try and claim only for Ash to nip in before him and Michael Toner to put the home side in front.
With the swirling wind and bobbly pitch causing problems for both sets of players, the conditions conspired to present Weston with two opportunities to extend their lead before the interval.
The first came when Bradley Watkins’ clearance from a Hickford back-pass went straight to Grubb, but his effort from distance didn’t have the required power to beat the scampering Aston Villa-loanee who’d just about retreated to tip the ball wide.
And then into injury time the wind caught a goal-kick of Luke Purnell and looped over the Clarets back line for Oliver Watkins to latch onto, only to be prevented by his namesake diving at his feet to clear.
With the wind behind them in the second half City went in search of an early opportunity for an equaliser having failed to test the Weston goalkeeper at all in the first period.
And their best chance by far presented itself when Rob Girdlestone’s strike from distance was blocked straight into the path of Cheek who, one-on-one with Purnell, stabbed the ball wide when you would have backed him to score.
Joe Ward was introduced in place of Rohdell Gordon and it almost coincided with City going 2-0 down as the youngster was shrugged off by Grubb in the centre of the field only for Oliver Watkins to slip at the vital moment to send his effort off target after being played in.
The Clarets’ two previous visits to the Woodspring Stadium had seen City fail to produce a goal and it looked like that would be on the cards again as they headed for their first defeat in four.
And with just under 10 minutes left in the game City’s fate was sealed after Grubb carried the ball to the edge of the area and riffled past Bradley Watkins and into the bottom corner to double the home side’s advantage.
As if to make certain of the points the home side extended their lead further in stoppage time after Joe McClennan tormented Girdlestone by cutting one way and then another allowing him to smash a cross across the face of goal for Ash to tap in.
City though did manage to prevent themselves going three straight visits to Weston without scoring as Mark Hughes was brought down in the penalty area deep into stoppage time. Lee Sawyer stepped up to send Purnell the wrong way but it was too little too late for the Clarets.
1. Luke Purnell
2. Pierce Mitchell
3. Ollie Barnes
4. Jamie Edge
5. Tom Jordan (C)
6. Clayton Fortune
7. Jake Mawford
8. Kane Ingram
9. Bradley Ash
10. Dayle Grubb
11. Oliver Watkins
12. Jake Buxton
14. Aaron Brown
15. Jacob Kane (On for Kane Ingram 57’)
16. Joe McClennan (On for Oliver Watkins 79’)
17. Syd Camper
1. Bradley Watkins
2. Rob Girdlestone
3. Michael Toner
4. Mark Hughes (C)
5. Harry Hickford
6. Marvin Ekpiteta
7. Lee Sawyer
8. Leon Redwood
9. Jaanai Gordon
10. Michael Cheek
11. Rohdell Gordon
12. Nicky Nicolau (On for Marvin Ekpiteta 79’)
14. Joe Ward (On for Rohdell Gordon 66’)
15. Christian Smith
16. Jake Gordon
19. Niklas Freund
Referee: Andrew Laver (Portsmouth)
Assistants: Alex Fisher (Portsmouth) Alex Blake (Waterlooville)
Weston-Super-Mare: Bradley Ash (19’, 90’), Dayle Grubb (80’)
Chelmsford City: Lee Sawyer (90+2’ (p))
Weston-Super-Mare: Kane Ingram (22’ – Foul)
Chelmsford City: Mark Hughes (15’ – Foul), Leon Redwood (33’ – Obstruction), Rob Girdlestone (50’ – Foul), Michael Cheek (77’ - Foul)
City Man of the Match: Lee Sawyer