Werder Bremen sink Sampdoria
Moldovan champions Sheriff Tiraspol, bidding to make the group stages for the first time, lost 1-0 at Swiss club Basel and Portugal's Braga, also yet to qualify for the first round proper, beat Sevilla by the same scoreline.
Anderlecht drew 2-2 at Partizan Belgrade, the only Serbian club to have played in the group stages, in Wednesday's other tie.
Enyeama gave one of the best goalkeeping performances at the 2010 World Cup when he restricted Argentina to a 1-0 win over Nigeria in the first round, repeatedly foiling dangerman Lionel Messi.
On Wednesday he showed a different side to his game as he calmly converted a third-minute penalty for the Israeli league champions in the less glamorous surroundings of Salzburg's half-empty stadium.
Salzburg, Austria's biggest-spending club but foiled in their last three attempts to reach the group stages, levelled with a long-range Nikola Pokrivac shot in the 28th minute.
Ben Sahar and Itay Shechter netted either side of halftime for Hapoel before Roman Wallner pulled one back for the hosts with a 66th-minute penalty.
Werder, who this week sold midfielder Mesut Ozil to Real Madrid, have never lost to Italian opposition at the Weserstadion and kept up their excellent record thanks to a second-half goal flurry.
Defender Clemens Fritz opened the scoring against Sampdoria with a superb left-footed strike, rifling home from the edge of the area in the 51st minute after a clearance landed at his feet.
Torsten Frings added a penalty in the 67th minute, awarded for a foul on Sebastian Proedl by Stefano Lucchini who was sent off as a result, and Peru forward Claudio Pizarro combined with Hugo Almeida to add the third goal two minutes later.
Sampdoria, who reached the European Cup final on their only previous appearance in the competition in 1992, grabbed an away goal through Giampaolo Pazzini in the 90th minute.
"We struggled to get into the game at first but in the second half we played well," said Werder coach Thomas Schaaf.
A 62nd-minute header by Matheus gave Braga their 1-0 win over Sevilla after the Spaniards had controlled the first half.
Valentin Stocker's close-range header earned Basel victory over Sheriff, who had Alexandr Erokhin sent off in the 86th minute before visiting goalkeeper Vladislav Stoyanov saved a late Benjamin Huggel penalty.
"I'm very happy with the result especially as we didn't concede a goal," said Basel coach Thorsten Fink.
In Belgrade, the four goals came in 13 second-half minutes as the game came to life after a cagey first period.