James Alabi netted for the Blues early on after taking Elliott Durrell's cross with a deft touch before turning and rifling home.
John Askey's men equalised against the run of play when right-back Andy Halls made a forward run and played a neat one-two before placing past Chester goalkeeper Alex Lynch.
Jack Mackreth's superb cross then allowed Anthony Dudley to put Macc ahead with a firm header.
Alabi hit the post in the second half, but Kane Richards did eventually stab Chester level from another Durrell ball in.
Norburn, however, took on Rhys Browne's ball and struck in low from outside the area with five minutes remaining.
Match report supplied by the Press Association.
Match ends, Chester FC 2, Macclesfield Town 3.
Second Half ends, Chester FC 2, Macclesfield Town 3.
Substitution, Macclesfield Town. Chris Holroyd replaces Anthony Dudley.
Ollie Norburn (Macclesfield Town) is shown the yellow card.
Goal!  Chester FC 2, Macclesfield Town 3. Ollie Norburn (Macclesfield Town).
Goal!  Chester FC 2, Macclesfield Town 2. Kane Richards (Chester FC).
Danny Whitaker (Macclesfield Town) is shown the yellow card.
Elliott Durrell (Chester FC) is shown the yellow card.
Substitution, Chester FC. Theo Vassell replaces Danny O'Brien.
Substitution, Chester FC. Kane Richards replaces Blaine Hudson.
Second Half begins Chester FC 1, Macclesfield Town 2.
First Half ends, Chester FC 1, Macclesfield Town 2.
Goal!  Chester FC 1, Macclesfield Town 2. Anthony Dudley (Macclesfield Town).
Goal!  Chester FC 1, Macclesfield Town 1. Andy Halls (Macclesfield Town).
Goal!  Chester FC 1, Macclesfield Town 0. James Alabi (Chester FC).
First Half begins.
Lineups are announced and players are warming up.

The 30-year-old, excluded from the World Championships by USA Swimming after a drink-drive conviction, won in one minute 54.75 seconds in Texas.
Phelps, an 18-time Olympic champion, also recorded the year's fastest times to win the 100m and 200m butterfly.
Ryan Lochte won world 200m individual medley gold in 1:55.81 on Thursday.
American Lochte, 31, also holds the world record for the event of 1:54.00, which he swam in 2011.
"Ryan and I have been pretty dominant in that race since we started racing back in 2004," said Phelps.
"I'm looking forward to getting back in and competing with him.
"I've been away from the sport for a while and I feel like I'm kind of back where I used to be. It's good to be here."
On Saturday, Phelps won the 100m butterfly in 50.45 seconds, faster than the 50.56 South African Chad Le Clos swam to win world gold on the same day.
A day earlier, Phelps won the American 200m butterfly title in one minute 52.94 seconds in San Antonio - 0.54secs faster than Hungary's Laszlo Cseh swam in winning the world title in Kazan, Russia.
Phelps, who was convicted of drink-driving in December and served a six-month ban from competition, now has a record 55 national titles.

Toilet roll, chocolate biscuits, orange juice and antibacterial wipes were all on the list of shrinking items reported by Which?.
The consumer group argued that making products smaller was a "sneaky way" of increasing prices.
It is further evidence of a trend towards shrinking products.
In January Unilever revealed that a number of ice creams it makes would be getting smaller.
The latest list from Which? includes:
Most brands, responding to the Which? research, said it was up to supermarkets to set the price. However, they would not disclose if they had charged the stores a lower wholesale price.
"Shrinking products can be a sneaky way of increasing prices. We want manufacturers and supermarkets to be upfront about shrinking products so consumers are not misled," said Which? editor Richard Headland.
Ollie Norburn grabbed a late winner as Macclesfield overcame Chester 3-2 at the Deva Stadium.
[NEXT_CONCEPT]
Olympic great Michael Phelps swam a third 2015 world-best time in as many days in the 200m individual medley at the USA National Championships.
[NEXT_CONCEPT]
A consumer group has revealed its latest list of shrinking supermarket items that still cost the same, or more.