Koscielny's goal was the second late victor in four Premier League games for Arsenal and allowed them to keep in touch with the league's top two, Spurs having beaten Manchester City earlier in the day to tighten the margins at the summit.
The Frenchman got on the end of Theo Walcott's flick on at Turf Moor in the third minute of stoppage time, touching the ball with his right boot before it hit his arm and went over the line.
"It was all about coming here and giving everything", he told Arsenal's official website.
When Arsenal were losing 4-3 to Liverpool at The Emirates, the club were still hammering out the details of signing Shkodran Mustafi from Valencia for £35m and Laurent Koscielny was on a break after his Euro 2016 exertions.
"We took it by the scruff of the neck and I think we played some really good stuff today, like I say we defended unbelievably well and if we keep playing like that we will pick up a lot of points".
Johann Berg Gudmundsson's header was turned away at full stretch by Petr Cech and Michael Keane headed a 75th-minute corner against the woodwork, although replays appeared to show the goalkeeper got his hand to it.
Arsene Wenger's team beat Burnley 1-0 with a brace so ugly and I mean ugly, that it has to be seen in order to be believed.
Sanchez went even closer inside the final 20 minutes, grazing the post with a close-range volley, before Arsenal's late pressure eventually told in a heart-breaking finale for Sean Dyche's side. "A win like this is important to us".
Dyche said it was a cruel way for his side to lose, but did not take the matter up with referee Craig Pawson after the match and took solace in his own side's performance.
"I want to give him back that, but I'm not the only example, there have been many".
"It's a big win because City lost", said Wenger, adding: "We were five points behind them going into this game, and when you get a chance to move closer you can't miss it, so we have to be pleased with the way we kept going to finally get the win".
"We know we're not market leaders, we're the minnows of the division, you can see by our spending power compared to everyone else". We made many technical mistakes today.