AC Milan had to settle for a 1-0 victory over Red Star Belgrade and there was a scare for 2005 Champions League winners Liverpool before they edged Maccabi Haifa 2-1 in Wednesday's third qualifying round action.
Ajax Amsterdam were 2-1 winners at FC Copenhagen, as were AEK Athens at 10-man Hearts after two late goals, and Hamburg SV drew 0-0 at home to Spain's Osasuna in other first leg games.
A month to the day since Italy won the World Cup, four of their squad -- Filippo Inzaghi, Andrea Pirlo, Gennaro Gattuso and Alberto Gilardino -- helped Milan to a morale-boosting win.
It was Inzaghi's 33rd birthday and the striker celebrated in style by poaching the match-winner after 22 minutes at San Siro, a rare moment of celebration for the club in recent weeks.
Docked points by an Italian sports tribunal for their part in the country's match-fixing scandal, Milan were only allowed to play in Europe by a highly reluctant -- and critical -- UEFA.
"The only disappointment is that we couldn't get another goal," said Milan coach Carlo Ancelotti.
"Given the stage of our preparation this season, I can't have any complaints about the performance. The lads showed a lot of grit and determination."
There was determination too at Anfield, where Liverpool went behind to Gustavo's 29th minute strike for the Israeli side but were spared
Welsh forward Craig Bellamy equalised four minutes later but the Reds looked to be heading for a surprise draw until Chile winger Mark Gonzalez rifled home the 88th minute winner.
"That's what we've bought them for -- to score important goals for us -- and they've done their job tonight," Liverpool captain Steven Gerrard told Sky Sports News.
"We'll be a better side come the second leg but there's still a job to be done," Gerrard added. "We need to work hard to get into the Champions League."
The second leg in two weeks' time will be at a neutral venue after UEFA ruled that the security situation in Israel meant no matches could be played there until further notice.
Klaas-Jan Huntelaar scored both goals for Ajax in Copenhagen as the Dutch side staked their claim to a place in the money-spinning group phase of the competition.
An 88th minute equaliser by AEK striker Pantelis Kapetanos and a stoppage-time own goal by Hearts defender Christophe Berra turned around their game in Edinburgh, as the Greek side conjured up a late victory.
Italian hopefuls Chievo Verona lost 2-0 at Levski Sofia, Valencia went down 1-0 in Salzburg, while Turkey was the place for goals with Galatasaray beating Mlada Boleslav 5-2.