As a result, Slutsky's men stay third in Group G, but move just one point behind the second-placed Swedes ahead of the final three qualifiers.
Below, relive the drama with our live commentary.
With Fabio Capello gone, all eyes are now on newly-appointed Russia coach Leonid Slutsky as he bids to rescue their Euro 2016 qualification campaign. Is it beyond rescue? Well, Capello certainly did his best to ensure they won't reach next summer's finals in France, but no, it can certainly be salvaged. But it will be hard. The road to redemption begins in Moscow as second-placed Sweden come to town in a must-win Group G game for the hosts. Nothing but a win will do.
If you have heard the name Slutsky before but can't quite remember where, he was the CSKA Moscow coach who only masterminded a 2-1 Champions League win at Manchester City last season. It may also be a Polish derogatory term for a promiscuous woman, but probably not. The 44-year-old takes the Russia job on the back of winning two league titles with CSKA in 2013 and 2014. He is no mug.
Anywho, let's whip out some team news before we have a greater look at Group G, shall we? I think so.
RUSSIA XI: Akinfeev; Agnashevich, V Berezutski, Smolnikov, Denisov; Dzagoev, Shirokov, Shatov, Zhirkov; Kokorin, Dzyuba
SWEDEN XI: Iasaksson; Bengtsson, Antonsson, Granqvist, Olsson; Forsberg, Seb Larsson, Ekdal, Wernbloom; Durmaz, Ibrahimovic
So, some things to note. First, Russia defender Sergey Ignashevich wins his 109th cap tonight, and he'll be honoured before kickoff. His first came against Sweden in 2002, in a game which saw a certain 20-year-old Zlatan Ibrahimovic score a last-minute equaliser. Zlatan is also back tonight of course, as is Pontus Wernbloom, who wins a first cap in three years. He played under Slutsky at CSKA. I wonder why Sweden boss Erik Hamren called him back?
So, Group G. It's tight. It's tighter than a scoutmaster's sheet bend. It's tighter than the half-ton man squeezing through a turnstiles. It's tighter than, well, you get the point. Second-placed Sweden have 12 points, four behind Austria and four clear of third-placed Russia with four games to play.
But Montenegro and Liechtenstein are only three points behind Slutsky's men. What does that mean? You guessed it - defeat against the Swedes tonight could see Montenegro or Liechtenstein - who play each other - move level on points with Russia should they win. And that would NOT be a good look for Russia. Nope.
In the words of a detective at Scotland Yard, should we have a look at the danger men? Why not and, not unlike Arsene Wenger at Arsenal, Russia don't have a lot of marksmen that they can all upon. With two goals, Artem Dzyuba is their leading goalscorer in this campaign - and one of those was a penalty. In fact, even more like the Gunners, Own Goal has scored two for them so far. Russia have technically scored nine, but they were awarded a 3-0 win thanks to Montenegro's fans following crowd trouble back in March so, really, they've scored six.
In contrast, Sweden have scored 10 - and Zlatan only has five of them. Indeed, the towering Paris Saint-Germain hitman has netted half of his country's goals in qualifying, and you just wouldn't back against him adding to that haul tonight. Turkey-born Erkan Zengin also has two under his belt.
Incidentally, this is the first birthday of the Otkrytie Arena. All together now - happy birthd.....no? Okay. It was precisely one year ago today that Spartak Moscow played its first game here - a friendly with Crevena Zvezda. There's a bit of pointless information for you. We're five minutes from kickoff.
Here come the anthems...
Oh god Mark Clattenburg is refereeing. This will be abandoned...
KICKOFF: And we're off in Moscow. The hosts, playing from left to right, get us underway. Let's go folks.
Slutsky looks a little like the Yellow King from the first series of True Detective. That's about the only thing I've observed in these opening minutes.
CHANCE! We very nearly saw the first goal. A long ball from Russia isn't dealt with by Granqvist, whose defensive header goes straight up in the air. It allows Artem Dzyuba to get in on goal, but he lets the ball bounce, before trying to salvage the chance with an overhead but it flies over the bar.
We haven't seen a whole lot of Ibra thus far, but the big Swede has just won an excellent knock-down for Durmaz, but he can't reach it and Russia clear. It's a very cagey encounter so far.
OFFSIDE GOAL! What a ball this is from Dzagoev, who splits Sweden's central defence as captain Shirokov runs through before finishing well, but it's correctly called offside. Good decision.
Very even game this. Both sides are going to score tonight. I'd put my house on it. Well, I don't own my house, but that's how confident I am.
CHANCE! Oh my word it could, and should, be 1-0 to Russia, as Aleksandr Kokorin gallops down the right hand side before biding his time and cutting back for the onrushing Oleg Shatov, who fires wide of the goal. Poor, poor finish that.
SAVE! It's all Russia now as their fans chant something in, eh, Russian. It's Kokorin again, who drives an excellent ball into the box and Shirokov's first-time snap shot is well saved by Isaksson. Sweden are needing to hang on here a little. Granqvist is having an absolute mare.
Ibra is such a handful up front. He's caused some real problems tonight. He's just so big and tall. That probably sounds like I'm in love with him.
We're midway through the first half then and still goalless. We've had chances, primarily from the Russians, but Sweden have had their moments. There will be goals.
CHANCE! Kokorin has been excellent in this first 25 minutes, but he should've done better. Zhirkov whips in a dangerous ball following a slick attack, but the reported Arsenal target miscues his header and it goes well wide.
It's gone a little scrappy in recent minutes to be fair. Russia's midfield is getting stronger as the minutes tick on, which spells bad news for Sweden because they can't get the ball.
You can see why Kokorin is reportedly wanted by Arsenal because he's looked a class apart in the opening half-hour or so. He's set up all of Russia's chances, but he also should have scored, or at least hit the target, with a wayward header a little earlier.
BOOKING: Dzagoev goes into the book for a foul on Ibrahimovic on the halfway line. He wanted that, the Swede. He held the ball up well before inviting a challenge from behind. He's wily, isn't he?
GOAL! Russia 1-0 Sweden (Dzyuba)
This is a brilliant goal. Igor Smolnikov squares to Shirokov, whose exquisite back heel puts through the right-back, but he pulls back for Dzyuba and the striker side foots home into an empty net - although he almost missed! Still, great work from Kokorin, Smolniok, Shirokov and Dzyuba. Six minutes until the break.
Sweden looked poised to finish the half strong, but I'm not sure they're going to score before the break. They look a little short on incentive. Victory isn't important for them as it is for Russia.
HALF TIME: Russia 1-0 Sweden
Well, that's it for one half, my friends. Artem Dzyuba's tap-in on 38 minutes is enough to earn a slender, but thoroughly deserved 1-0 half-time lead in Leonid Slutsky's first game as Russia coach. They've been much better overall, fashioning the much better chances, but it's only 1-0 and Sweden, spearheaded by the always-dangerous Ibrahimovic, will look to come back strong in the second half period. Don't write them off.
RESTART! And we're back. Some bad news for Ibra, who has been hauled off at half time for Ola Toivonen. Big, big ask for Sweden now, with Ibrahimovic having scored half of their 10 goals in Euro 2016 qualifying thus far.
Brilliant goalkeeping from Akinfeev, who races off his line to take out Forsberg, who was baring down on goal after getting the better of Berezutsky, but the keeper took the ball before the man. Excellent.
Despite the absence of Zlatan, Sweden have started the half well. It's all them at the moment, and Toivonen just miscued a diving header a few moments ago. Still, it's a big ask for them to get anything out of this. Ibrahimovic is their go-to-guy, isn't he.
BOOKING: Wernbloom, on his return to national team football after a three-year absence, goes into the book for a scissor tackle on Kokorin. Might be fortunate to stay on. It was rash. It was so rash that Kokorin needs a dollop of Sudocream on his ankle.
We said Leonid Slutsky looked like the Yellow King from True Detective, but actually, he looks like Action Bronson without the beard. And the weight. And the swag.
We said Sweden started the half well, and they did, but it's all Russia now. I wouldn't say a goal is imminent, but you wouldn't be surprised.
Oh god what an opportunity this is. Russia have had immense joy down Sweden's left-hand side tonight, and right-back Smolnikov runs on to a lofted ball. He's got three red - or whatever colour Russia are playing in - shirts in the middle, but his final ball is dreadful and goes well over the heads of all three. Seriously what colour is that? It's like red wine mixed with blood. Sorry to lower the tone.
Right, there's slightly less than half-an-hour left and Russia, it has to be said, still looking comfortable - but they need a second goal to be sure.
CHANCE! Boy do they need a second goal! Sweden fashion their best chance of the match when Emil Forsberg cuts inside before firing at goal, but Akinfeev pulls off an excellent save at his near post, before Forsberg thrashes into the side netting with the rebound. Sweden asking questions.
SHOT! Back come Russia in their pursuit of what would surely be a killer goal, and you could park a jumbo jet in Sweden's midfield as Roman Shirokov runs through before firing at goal, but Isaksson makes a comfortable save. Probably should've done better.
Kokorin really has been excellent in this game. He's been fouled about 50 times (probably four times at most) and it's Olsson this time, but Zhirokov's resulting free kick is poor and glides out for a goal kick.
SUB: Russia make their first sub now as Zhirokov makes way for Oleg Kuzmin
Just a reminder that, as it stands, Russia stay third, but they move within a point of second-placed Sweden. Group G leaders Austria can effectively book their spot in France next summer if they beat Moldova in Vienna later on tonight, which they surely will. It's really a battle for a second place and Russia, if they win this, are right back in that battle.
Right, 12 minutes to go and the Russians still just about ahead. Sweden are getting better, although they've only had, like, one proper chance in the whole contest. If you've just joined us then better late than never, and Ibrahimovic left the field at the break following injury. Toivonen replaced him and, well, he's not fit to tie his man bun, is he? Nope.
Russia just starting to run out of legs now as Sweden continue to threaten. They've had most of the possession in this half, but they're doing precious little with it at the moment. Igor Smolnikov is on the ground now, probably in a bid to waste a few minutes, but who knows - maybe he's genuinely injured.
SUB: Russia make their second sub now as Aleksei Ionov replaces goalscorer Dzyuba for the last 10 minutes or so
OFF THE LINE! Huge from Olsson, as Kokorin runs through on goal at an angle before poking past Isaksson, but the shot didn't have enough pace and the Sweden left-back makes the clearance. Massive. Colossus. Gigantic.
Strangely, Russia finishing much better, despite being very, very tired, but there's still five minutes to go. Don't rule anything out.
SAVE! What a save from Akinfeev as he turns Marcus Berg's ferociously-struck shot over the bar, with the corner coming to nothing. He's not had much to do in this game, but the Russia stopper has pulled off two huge saves in this second half.
SAVE! And again from Akinfeev - and it's only Berg he denies again! The second-half sub looks to prod home from close range, but a lack of clean contact, coupled with Akinfeev's alertness to race off his line, sees the Russian stopper save again. Huge.
FOUR MINUTES ADDED ON:
CHANCE! If Russia don't win then Kokorin might have to emigrate. He's just missed a gilt-edged chance by volleying over the bar from inside the box, with just Isaksson to beat!
FULL TIME: Russia 1-0 Sweden
And that's it! Dzyuba's winner after 38 minutes is enough to give Slutsky a winning start but, more important, also breathe new life into Russia's Euro 2016 qualification campaign. They stay third, but they move one point behind the Swedes in second place. Forget about Austria, who can effectively seal their spot in France if they beat Montenegro later on. Anywho, that's all we've got time for. Thanks for joining our live text commentary and we'll see you next time. Cheers.