And Mansfield also registered their first clean-sheet in 12 league matches - but only after two injury-time scares.
First Ville Vaisanen's powerful header from a late free-kick was spectacularly saved by the diving Jason White.
Then, in the 93rd minute, Gaetano Giallanza, unmarked at the far post, somehow struck substitute Alun Armstrong's deep cross wide.
The Stags, victorious in just one of their previous four league matches at Field Mill, were forced into changes after injuries to full-back John Mullins and midfielder Michael Boulding.
On-loan Leicester defender Alan Sheehan returned, Gareth Jelleyman switched to midfield, Reet got his first home start of the season and former Nottingham Forest defender Jon Olav Hjelde returned after a nine-match absence through injury.
Darlington, unchanged from Saturday's draw with Shrewsbury, again left six-goal striker Barry Conlon on the bench.
And in the first half they missed his goal threat, despite dominating the match with neat, quick passing that often ended without a goalscoring chance.
After just three minutes a mistake by Alex John-Baptiste allowed Michael Cummins a shot which Sheehan vitally charged down. Then Darren Holloway's drive was saved, before Kalusivikako Ngoma fired over.
The best chance of the half fell to Julian Joachim, the pacy striker racing clear on to Cummins' clever pass only for White to cling on to his shot.
The Stags' re-shaped line-up also created little. Stephen Dawson's long-range drive forced Sam Russell into a full length diving save, Matt Hamshaw's shot from an incisive Jelleyman break was blocked and, in injury-time, Richie Barker's fierce drive ricocheted to safety off Holloway.
The second half started off in similar style, with little goalmouth action despite the half-time introduction of experienced Darlington striker Armstrong for the ineffective Joachim.
But the game finally came alive on the hour when Sheehan and Jelleyman combined excellently on the left wing for Sheehan to cross to Reet, who turned superbly and held off two defenders before firing across keeper Russell and into the bottom corner.
Substitute Armstrong almost found an immediate equaliser, but his clever back-heeled finish went straight to White.
Johnson's long-range effort then forced a fumble from home keeper White, while at the other end substitute Giles Coke wasted a clear chance to make certain of the points when he dragged his 18-yard shot well wide.