For 65 minutes the Blues produced more than enough to suggest an unlikely result against the promotion-chasing Reds was possible.
With more of the ball and more of the chances, the home side were the better team but without loan star Matty Taylor - now back at Forest Green Rovers - the goal threat has evaporated.
Alfreton took the lead with a penalty with 25 minutes left and fussy referee Darren England could probably have blown for full time then.
Chester huffed and puffed in the closing stages and nobody can question the effort or commitment of the players but the lack of quality up top was all too evident.
Burr made one change in response to the uninspiring loss at relegation rivals Aldershot Town on Tuesday, dropping Craig Lindifeld and recalling Matty Brown on his 24th birthday.
Brown - on loan from Chesterfield - had missed the last three games with an ankle knock and his return meant a formation change to 4-4-1-1 and a positional reshuffle.
Burr moved Ross Killock to right back, pushed Lewis Turner into right midfield and fielded John Rooney in the hole behind Gareth Seddon. New signing Craig Hobson, who joined from Skrill North club Guiseley AFC, was named on the bench.
Alfreton came into the game in third place having consigned Hyde to relegation with a 3-0 win midweek. Reds boss Nicky Law replaced left back Connor Franklin with Bradley Wood and named ex-Blues goalkeeper Jon Worsnop in his side.
Chester made a solid enough start and Rooney opened up the home defence on seven minutes with a quick free kick but Seddon couldn't quite connect with his pass.
Rooney was involved in a promising move on 10 minutes but Killock sent his cross high and wide with team-mates queuing up in the area.
Lewis Turner and Nathan Turner created the first real chance five minutes later, linking well down the right and the latter pulled back to the busy Rooney but his low shot was well saved by Worsnop.
The home side suffered a blow on 19 minutes when Killock pulled his hamstring and had to be replaced. Burr opted to bring on Hobson for his debut with Lewis Turner reverting to right back.
Chester continued to enjoy the greater possession and Brown went close from a Rooney corner on 25 minutes but his goal-bound header was blocked.
Alfreton came back into the game with burly striker John Akinde forcing Aaron Chapman into a stop on 32 minutes with an effort from distance and Shaun Harrad couldn't get hold of his shot after getting the wrong side of the defence.
Chapman denied Akinde again three minutes later, getting down well to hold on to the big forward's low shot from inside the area.
Andy Bond showed some good touches in the middle for the home side and his smart pass released Mahon down the left but the winger saw his cross blocked.
With the half time whistle about to sound, defender Michael Wylde clattered into Rooney on the edge of the box. Referee Mr England gave a free kick but there was greater concern for Wylde who suffered a suspected fracture tibia on landing.
Wylde left the field on a stretcher although Mr England showed him a yellow card for his foul and Rooney took the free kick but his effort hit the wall as the sides went into the break level.
Chester came out with intent in the second half and had a fine chance on 47 minutes when Rooney released Seddon with a delicate chip but the experienced striker could only drag his shot wide.
Alfreton should have taken the lead a minute later when the impressive Godfrey Poku broke quickly and released Akinde but Chapman made a good stop with his legs.
It was the home side who were next to threaten when Worsnop made a mess of dealing with a Joe Heath cross and birthday boy Brown fired over under pressure.
Lewis Turner put a trademark teasing cross into the box, Mahon produced a good ball from the other flank and Hobson clipped one into a dangerous area as the Blues pressed but were lacking that killer touch.
But those familiar defensive fragilities nearly cost the hosts when Chapman produced a smart stop to deny Harrad and the ball fell for Akinde but he shot wide from six yards out.
Chester were looking more and more open to the counter and a break down the right resulted in a penalty for the visitors on 65 minutes.
Daniel Bradley fired a low cross into the six yard box and Akinde got to the ball first and poked wide but referee Mr England ruled Chapman had fouled him.
It was a debatable decision but Harrad stepped up and rolled into the ball into the bottom corner to put the Reds ahead against the run of play.
Burr introduced Jamie Menagh and Brendon Daniels for Nathan Turner and Seddon as he went in search of a leveller but the home side were fast running out of ideas.
Alfreton looked dangerous on the break and the hosts began to launch balls forward in hope more than expectation.
Without Seddon leading the line the home side lacked a focal point in attack and the visitors were never troubled in the final 10 minutes to leave the Blues teetering above the bottom four.
Chester FC: Chapman, Killock (Hobson 19), Brown, Linwood, Heath, L Turner, Bond, N Turner (Menagh 72), Mahon, Rooney, Seddon (Daniels 75).
Subs: Danby, Bridge.
Booked: Mahon, Chapman, Daniels.
Alfreton Town: Worsnop, Law, Wylde (Kempson 45), Fenton, Wood, Jones (Meadows 56), Bradley, Poku, Shaw, Harrad, Akinde (Clayton 75).
Subs: McFadzean, Andrews.
Goal: Harrad 65 (pen).
Booked: Wylde, Akinde, Law, Harrad.
Referee: Darren England (Barnsley)
Attendance: 2,042.
