Grzegorz Krychowiak netted the only goal two minutes after his team mate Christopher Glombard was sent off on the hour and they held on for victory with even second-half substitute Beckham failing to lift the visitors.
PSG, who host Valencia in a last 16, second-leg Champions League match on Wednesday having won 2-1 at the Mestalla, looked nothing like the side who handed out two 2-0 defeats to arch-rivals Marseille in Ligue 1 and the French Cup in the past week.
Saturday's result sets the stage for a nail-biting run-in to the Ligue 1 season as second-placed Lyon can move level on points with PSG by winning at Stade Brest on Sunday.
PSG have 54 points from 27 matches with Lyon on 51 from 26. Reims eased their relegation fears as they moved up to 16th on 27 points, three points above the safety zone.
St Etienne thrashed 10-man Nice 4-0 to jump into the third Champions league qualifying spot on 47 points.
Pierre-Emerick Aubameyang netted an early penalty before Valentin Eysseric was sent off in the 24th minute for a tackle on Jeremy Clement, who went to hospital with a fractured ankle, according to St Etienne.
Les Verts had the game their own way after that but had to wait until the hour mark before Brandao got the first of his two goals either side of a Yohan Mollo strike.
Nice slipped one place to fifth with 45 points, a point behind Marseille who can reclaim third spot from St Etienne by beating Troyes on Sunday.
Tempers boiled over in the Coriscan derby with five players sent off as Bastia beat fellow strugglers Ajaccio 1-0.
PSG like to play on the counter-attack, even against the lower placed Ligue 1 teams, but the tactic does not always produce dividends as they chase a first league title since 1994.
They had their chances, with Ibrahimovic's fierce shot from just outside the box in the 13th minute blocked by goalkeeper Kossi Agassa after the Swede was set up by Blaise Matuidi.
However, it was Reims who came closest to opening the scoring when Anthony Weber's 34th minute effort was cleared off the line by Marco Verratti.
PSG's resulting counter-attack saw Ezequiel Lavezzi through one-on-one with Agassa but he missed the target.
Reims's Gaaetan Courtet then skied a header over the bar from a cross by Odair Fortes.
Glombard was sent off after getting two yellow cards in 14 minutes but the home side continued to press and were rewarded when Krychowiak latched on to Mohamed Fofana's downward header.
Beckham, in his third PSG appearance since joining as a free agent on January 31, replaced Verratti four minutes later but he was unable to inspire the lacklustre visitors.
The Corsican derby was a hot-headed affair as Ajaccio, who had lost Matthieu Chalme to a second booking 10 minutes from time, were reduced to nine men when Dennis Oliech was dismissed in added time for fouling Bastia keeper Mickael Landreau.
In the brawl that followed Bastia's had Florian Thauvin and Jerome Rothen sent off with Ajaccio's Benjamin Andre.
In the relegation battle, Nancy kept alive their hopes of staying up with a 2-1 win at Sochaux to move off the bottom, a point above Troyes and six points from the safety zone.