After falling behind in the early stages, York bravely challenged Storm Dollar's conditions and qualified for the fourth round of the FA Trophy, which they won in 2012 and 2017.
Matty Cornish took advantage of a mistake at the back to score early on, slotting past Harrison Mehle with little effort.
Thankfully, City responded quickly when David Ajiboye ran through a one-on-one and lofted the ball past Pete Jameson to score his first goal for the club.
The score remained 1-1 until half-time, when centre-back Ricky Aguiar's long-range shot found the bottom corner of the goal.
Ajiboye set up Sinclair in the six-yard area and pushed home, and the Minsterman confirmed his progress with the trophy.
It was a welcome respite from league play for York, who were guaranteed to remain at the top of the National League as their FA Trophy campaign began.
The team they visited was Darlington, managed by former City manager Steve Watson, who sit 13th in the National League North with just one loss in their last seven games after 19 games.
The main talking point during Darlington's visit to the LNER Community Stadium was that former City side Quakers currently have eight players on their roster, with a number of familiar faces returning to York.
Of those eight, six featured in the matchday squad and five started, including York stalwarts Pete Jameson and Scott Barrow.
Cedric Mayne, Will Hatfield and Callum Griffiths also started for the visitors, with Watson naming Tom Platt as a sub.
One of Hinshelwood's focuses in this week's pre-match press conference was that winning the FA Trophy is among his priorities this season, especially as York are currently well within the race for promotion from the National League. The question was where to put it.
But Hinshelwood said the FA Trophy was a throwaway game for him and denied any suggestion that the manager was aiming for a Wembley visit.
With that in mind, York manager has named a strong squad for this afternoon's game, making four changes from Saturday's 6-2 win over Maidenhead United.
Jeff King started for the first time, with David Ajiboye, Dipo Akinyemi and Ricky Aguiar all included.
Olly Pearce, Marvin Armstrong, Alex Hunt and Malachi Fagan-Wolcott all withdrew, with the latter two not included in the matchday squad.
Their absences created opportunities for substitutes Paddy McLaughlin and Mo Fadella, as well as Billy Chadwick, who returned from a loan spell at Boston United.
The weather would have a clear impact on the match and it was initially difficult for York's usually patient passing play to find a breakthrough against a solid Darlo side.
It was the visitors who had the chance to take the lead, Cornish's header being saved by Marais from close range, but an injury to Adam Crooks forced an early substitution on City's centre-backs.
Darlington took the lead in the 14th minute when Cornish intercepted a loose pass across the backline and fired a calm finish past Male, but he was not to blame.
But York responded in time when Ashley Nathaniel-George played an impressive through ball to Ajiboye, who fired a precise finish past Jameson.
City looked to score a second goal but were repeatedly thwarted by Jameson, who first thwarted Sinclair's attack but Akinyemi was denied with a header from seven yards.
Akinyemi regretted another missed opportunity. A pinpoint cross from King found the striker in a good position, but his volley hit the post.
Callum Howe's header went just over Darlington's crossbar, while Sinclair's free-kick attempt went wide of Jameson's near post.
Darlington got off to a strong start in the second 45 minutes as strikes from Mayne, Griffiths and Kayden Kelly went wide.
However, York were able to weather the storm from Darlington and Darragh and their corner kick was cleared by Agia on the edge of the area.
He opened up and landed a nice blow into the bottom right corner that rooted Jameson to the spot.
Just four minutes later, the match was over.
A wonderful pass from the Minstermen saw Ajiboye get in behind, and he selflessly met the onrushing Sinclair and side-footed home from just a few yards in front of goal.
Marais was called upon to play in the closing stages, first knocking down Platt's shot and then blocking another Cornwall shot into the back of the net.
York are currently in the fourth round of the FA Trophy and despite Darlington's better results, they have maintained their excellent home record and are on a winning streak.
York City: Male 7, King 6, Howe 7, Crooks (Armstrong 6, 11), Felix 7, Batty 7, Aguiar 8, Nathaniel-George 7 (Chadwick 6, 70), Sinclair 7 (Fadela, 82), Akinyemi 5 (Pierce 6, 61), Ajiboye 8 (McLaughlin, 82)
Subwoofer not used: Watson, Thomas
Goals: Ajboye (15), Aguiar (69), Sinclair (71)
Yellow Card: Battery (60)
Darlington: Jameson, Griffiths, Leeds, Forbes, Barrow, Cornish (Lambert, 60), Hatfield (Rivers, 86), Flint (Pratt, 75), Kelly, Salkeld (Maskell, 60), Main (Al-Shabib) , 86 years old)
Subs not used: Hedley, McGrath.
Target: Cornish (13)
York's star man: Ricky Aguiar. A really impressive shift in an unknown position, and a great goal.
Referee: Andrew Miller
Number of attendees: 3316 (751)
Number of shots (on target): 11 (7) | 10(5)
Corner: 4 | 3
Fouls: 8 | 8
Offside: 4 | 2
