Michael Spillane opened the scoring with an impressive finish from his own half. Rhys Murphy bundled the ball over the line to add a second to City’s tally late in the first half.
Tom Wraight rocketed a shot into the top corner midway through the second half, but the Stones fought back and almost upset the Clarets.
The hosts were handed a lifeline when Max Porter deflected the ball past his own goalkeeper on 51 minutes. Substitute Stefan Brown revitalised the Stones by heading beyond Nathan McDonald fifteen minutes later.
Porter was harshly penalised when the ball deflected off his boot onto his arm, but McDonald produced a low save to his left to deny Danny Green from twelve yards as City retained all three points.
Amos Nasha and Jonny Giles returned to the starting eleven as City boss Rod Stringer looked to make changes from the side that drew 0-0 with Welling United last weekend. Defender Rob Swaine missed out through injury and Scott Fenwick could only take a spot on the bench after taking a knock last week. Jerry Amoo made his first appearance in the squad since signing, filling one of the substitute places.
The Clarets started the game the brightest of the two sides, forcing their way forward down the wings. Matt Young picked out Wraight inside the box, but his header failed to trouble Jonathan North as it flashed wide inside ten minutes.
Bobby Wilkinson’s side thought they had the lead on sixteen minutes as David Pratt headed Bradley Hudson-Odoi’s left-wing cross past McDonald, but the flag was raised for offside.
Wadah Ahmidi looked to test the City defence from range but they stood firm and blocked the tame effort.
Moments later, a sublime spell of play from Spillane saw the defender open the scoring on 27 minutes. After winning the ball just outside his own box, the fan-favourite drove forward, spotted North off his line and lobbed the Stones ‘keeper from 65 yards.
Following on from Spillane’s opener the Clarets looked to extend their lead. Wraight sliced an effort over from close range but moments later Murphy doubled the visitors’ lead, bundling home Young’s low cross from the right.
Nasha’s driving run sent City on their way to find a third before the interval, but the defensive midfielder was chopped down by Freddie Grant and the Stones man received a yellow card.
The Stones started the second half brightly as they began piling pressure onto the City defence, but the Clarets’ back-line stood firm and resolute.
Things progressed from good to excellent when Anthony Church picked out Wraight, with the winger instinctively lashing home from 25 yards as he left North flat-footed to increase City’s lead to three.
But, two minutes later, the hosts had a lifeline. Defender Rhys Tyler curled a cross into the box, City failed to clear and the Stones scored; a deflection off Porter took the ball past McDonald.
The Stones should have reduced the deficit further when a floating cross picked out Pratt inside the box, but the hosts’ man was unable to convert from close range and a minute later Green fired over from 18 yards.
With the Stones in search for a second, Wilkinson made his first change of the game: replacing Pratt with Brown. Meanwhile Stringer replaced the tiring Nasha with Taylor Miles on 60 minutes.
Hudson-Odoi poked wide from ten yards after good movement from Bradley Bubb created space, then Brown dragged wide from the edge of the box. But the substitute made amends for his earlier miss as he latched onto Grant’s knock down to fire past McDonald just past the hour-mark.
The hosts were awarded an excellent chance to level the game on 70 minutes when Porter was penalised for handball, but McDonald dived to his left and kept out Green’s low strike. Wealdstone came close from the resultant corner, but a City defender appeared to smash the ball off the line.
Stringer made his second change just after the spot-kick, replacing Wraight with Chris Whelpdale. The substitute should have wrapped up proceedings when he connected with Murphy’s left-wing cross, but Whelpdale’s diving header sent the ball marginally wide.
Both sides made further changes as Scott Fenwick replaced Porter for City, while Matthew Lench and Jeffrey Monakana replaced Hudson-Odoi and Ahmidi for the hosts.
Bubb had two chances late on to equalise, striking the post as he found himself one-on-one with McDonald before heading wide moments later, but the Clarets held on to secure a hard-fought set of points.