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Two goals in the final 20 minutes ended Dartford's season as Chelmsford progressed to next week's final on Sunday.
The Darts never did enough in front of goal during the two games, although the tie remained evenly balanced until the latter stages.
In the end, though, it was Chelmsford who created the most chances and deserved to win the National League South play-off semi-final second leg.
Dartford made two changes from the first-leg stalemate with Danny Harris coming in at right-back and Andy Pugh starting up front. Keaton Wood and Lee Noble both dropped to the bench.
It was not until the latter stages of the half, however, that Dartford started to get into their stride.
Hayes' 34th-minute free-kick was met by the head of Ronnie Vint but his goalbound effort hit a visiting defender. Harris and Elliot Bradbrook both looked tried to convert the loose ball but the post and Bush somehow prevented Darts going ahead.
Chelmsford's Max Porter was booked for a foul on Ronnie Vint before the visitors nearly grabbed he initiative but Deren Ibrahim did well to keep out Robbie Willmott's swerving long-range free-kick.
The visitors replaced Luke Daley with Bagasan Graham at half-time.
It was all-Chelmsford after the break as first Bush fired low against the base of the upright and then Chris Dickson headed over Bush's long throw-in.
Dartford lost Alex Brown to injury but the fact it took them until the 68th minute to have their first shot on goal over the two games spoke volumes about the quality in the final third.
Hayes' deep cross was headed back at the far post by Luke Wanadio and Bradbrook's volley was comfortably saved by Ross Fitzsimons.
The crucial first goal came with 20 minutes remaining. Tom Bonner dived in near the corner flag when he should have held back and Dickson's cross matched his approach work, giving Louie Theophanous a close-range header he couldn't miss.
Darts made a double change to get back into the game but they could not keep the ball in Chelmsford's half for any period of time.
It was always the visitors who looked likely to score a second. Dickson broke and crossed for Mullings and Josh Rees, who were both unable to convert.
The second time Dickson was afforded space he made Dartford pay by scoring the second goal to send his team through with five minutes left. Cutting in from the left, the former Charlton striker produced a clinical finish beyond Ibrahim to send the visiting fans wild.
Dartford claimed a consolation deep into stoppage time when Ofori-Acheampong scored from just inside the penalty area but by then it was too little, too late.
There were unsavoury scenes after the final whistle when Bush taunted the home fans, who had given him stick during the two games, and a home supporter ran onto the pitch to confront him.
Dartford: Ibrahim, Harris, Wynter, Bonner, Vint, Brown (Noble 55mins), Hayes, Bradbrook, Ofori-Acheampong, Pugh (Pavey 77mins), Wanadio (Murphy 77mins). Subs not used: Wood, Wells.
Chelmsford: Fitzsimons, Hill, Daley (Graham 46mins), Porter, Bush, Haines, Willmott, Rees, Theophanous (Haysman 86mins), Dickson, Jeffers. Subs not used: Miller, Cornhill, Johnson.
Attendance: 2,622.