Guy Hollis was dismissed in the first half and Concord took advantage, breaking the deadlock before half time through Lee Minshull and then extending their lead in the second half through Alex Wall.
But Slough stayed in the game and Francis Amartey pulled one back ten minutes from time, before Togwell bundled the ball in from a late corner kick to rescue a hard-earned point.
Chris Flood, playing his final match for the Rebels, was named in the starting line-up alongside Warren Harris, who was pushed up into the attack.
Both sides threatened in the early exchanges, with Concord keeper Sam Beasant saving one-on-one against Warren Harris, and Jack Turner keeping out Juan Luque’s effort at the other end.
Slough had a let-off when Concord debutant Taylor Mahoney connected with Alex Wall’s cross, but failed to find the finish.
The controversial moment of the match came with 35 minutes played, when Hollis was shown a straight red card after tussling with Concord striker Wall. After some deliberation and discussions with his assistant, referee Dean Skipper adjudged that Hollis was the last man, and that the Slough centre-back had dragged Wall to the ground when he had a clear path to goal, resulting in the dismissal.
Concord looked to take advantage of their numerical superiority along with the strong wind behind them for the remaining ten minutes of the half. They did so on 44 minutes, with a deep cross from Luque being met by captain Minshull, who headed in the opening goal.
Half time came with lots for Jon Underwood and Trent Phillips to think about, with Neil Baker unavailable on the day. After initially moving to a 4-4-1 formation after the sending-off, the Rebels started the second half in a 4-3-2 looking to get back into the game.
However, Slough found themselves two goals down in short order, as Wall was able to tap in from six yards when a cross drifted over the heads of Mark Nisbet and Sam Togwell.
At this point it looked as if it may be a case of keeping the score down, but Slough used their bench in order to try to get back into the match, with Matt Lench replacing Billy Clifford and Francis Amartey replacing Warren Harris to offer something different in attack.
The two substitutes made an immediate impact, with Lench driving the team forward and Amartey able to use his pace and strength to reach through balls ahead of the defenders.
For a moment Concord thought they had a killer third goal, but Jack Midson was ruled offside and the goal was chalked off.
The belief was still there amongst the Slough players that a positive result was still possible, and with eleven minutes remaining a magnificent individual goal from Amartey reduced the deficit to just one.
A clearance upfield from Wells was allowed to bounce by the Concord defence, and Amartey seized control, heading the ball past one defender before flicking it over a second. With Beasant left to beat, Amartey smartly lobbed the ball over the keeper’s head, with the ball just squeezing inside the post – an excellent first goal for the club.
The Slough fans found their voices again, and the players responded by throwing everything they had at getting an equaliser.
Josh Jackman was inches away from getting that equaliser after Lench’s cross was headed out to the edge of the box, however his attempt was tipped over the bar by Beasant.
Time was running out, but a Slough corner kick two minutes into stoppage time bore fruit. Lee Togwell threw himself at Lench’s delivery, getting his head to it just in front of a defender. Beasant stretched out a leg to push the ball onto the post but it rebounded back towards him and over the line to send everyone in Slough colours into wild celebration.
Report: Will Tuck
Concord Rangers Lineup
Sam Beasant, Jed Smith, Sam Ashford (Aron Pollock 66), Connor Essam, Lee Minshull (c), Tyrone Sterling, Juan Luque, Sam Blackman, Alex Wall, Jack Midson (Danny Green 71), Taylor Maloney (Adam Topley 75). Unused substitutes: Joe Gardner, Josh Glover.