In the Albanian city of Shkodra, the hosts missed a penalty but still outplayed Russia with goals by Altin Rraklli, Altin Lala and Igli Tare. Dimitri Loskov replied for the Russians.
Albania now has four points and Ireland three.
In other Euro 2004 qualification action Saturday, defending champion France crushed Malta 6-0 to move eight points clear in its group and looks to be winning the race to be first qualifier for next year's finals in Portugal.
Italy, runner-up in 2000, improved its qualifying hopes with a 2-0 victory over Finland, while in the top game of the night, the Netherlands and the Czech Republic tied 1-1.
England gained vital points but gained few friends, and maybe more critics, after an unimpressive 2-0 beating of soccer minnow Liechtenstein.
Before a sellout 3,500 fans in the Liechtenstein capital Vaduz, Michael Owen headed the first and David Beckham struck home a trademark free kick.
But Sven-Goran Eriksson's team missed more chances and came close to conceding goals in the last few minutes against one of soccer's weakest teams, which has just one victory in 40 competitive matches.
By contrast, Ryan Giggs scored one and set up two as Wales stayed top of its group with a 4-0 beating of Azerbaijan.
Spain was held to a 2-2 tie by Ukraine thanks to a last minute equalizer, and Denmark scored a 5-2 victory at Romania whose coach, Anghel Iordanescu, quit.
There was crowd trouble at Georgia's 2-1 home loss to Ireland in Tbilisi, where home fans threw an open penknife and a bottle onto the field.
Thierry Henry and Zinedine Zidane each scored twice in Lens as France made it four wins in four games at the top of Group 1.
Sylvain Wiltord and David Trezeguet added the other goals as the French moved eight points ahead of Israel and Cyprus who tied 1-1 at Limmasol.
Manchester United star Giggs inspired Wales to a fourth Group 9 win in a row as Mark Hughes' team moved closer to its first European Championship appearance since 1976.
Giggs set up a goal after just 13 seconds for Newcastle's Craig Bellamy, provided a cross for John Hartson's third just before halftime and then ran clear of the Azerbaijan defense for the fourth in the 52nd. Gary Speed netted the second in the 40th.
Returning to the Italian lineup for the first time in nine months, Francesco Totti set up two goals for Christian Vieri in a 2-0 win over the Finns at Palermo to climb to second and stay five points behind the Welsh.
The Italians jumped above the Serbs, who aren't playing in this latest round.
Ruud van Nistelrooy became the third Manchester United player -- as well as Beckham and Giggs -- to score Saturday when he put the Dutch ahead just before halftime against the Czechs in Rotterdam.
But Borussia Dortmund's towering striker, Jan Koller, equalized in the 67th minute, and the two teams are tied on seven points at the top of Group 3 above Austria, which didn't have a game.
Spain was on the brink of going four points clear at the top of Group 6 until Oleksandr Horoshkov volleyed a last minute equalizer for Ukraine in Kiev.
After Andriy Voronin had fired the home team ahead in the 11th minute, three goals came in the final six. Real Madrid star Raul Gonzalez and Joseba Etxeberria put Spain ahead until Horoshkov snatched a tie, which means Ukraine remains one point behind.
Scotland strengthened its chances of qualifying for a big championship for the first time in five years by beating Iceland 2-1 at Hampden Park, a result which gives the visitor little chance of making it to next year's finals in Portugal.
Although Chelsea striker Eidur Gudjohnsen canceled out a 12th minute strike by Kenny Miller, Dundee defender Lee Wilkie headed the winner for Berti Vogts team with 19 minutes remaining.
Scotland now has seven points from three Group 5 games and would have been in an even better position had it had it not dropped two against modest Faeroe Islands in a disappointing 2-2 tie.
Dennis Rommedahl scored twice as Denmark won 5-2 in Romania to go to the top of Group 2 after twice falling behind in Bucharest.
Adrian Mutu twice put the Romanians in front only for Rommedahl, Thomas Gravesen, Jon Dahl Tomasson and an own goal by Cosmin Contra to give the Danes all three points.
With leader Norway idle until Wednesday's match against Luxembourg, the Danes lead on goal difference, the Romanians dropping to third.
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