Fenerbahce clung on for a 2-2 draw away to Young Boys Berne despite playing the entire second half with 10 men and Portugal's Braga beat Celtic 3-0 in other third qualifying round, first leg ties.
Suarez, who scored 35 league goals for Ajax last season, needed only 13 minutes to make his mark when he gave the four-times European champions an early lead against the Greeks.
Ajax failed to capitalise on their early breakthrough, however and Vladimir Ivic levelled 18 minutes from time to put PAOK in the driving seat for the second leg next week.
Suarez scored three goals at the World Cup where he helped Uruguay reach the semi-finals but was also involved in the controversial handball incident in the quarter-final against Ghana when he stopped a goal-bound shot on the line and was sent off in the last minute of extra-time.
Suarez was seen celebrating as Ghana missed the resulting penalty - which would have given them a 2-1 win - and Uruguay went on to win the penalty shootout.
Group stage regulars Fenerbahce twice led in Switzerland but were denied victory by a late penalty.
Emre Belozoglu gave the Turks a fifth minute lead but Argentine defender Emiliano Dudar headed Young Boys level in the 18th minute.
The hosts then took control, Moreno Costanzo hitting the post, but were stunned were Fenerbahce regained the lead just before halftime with a long-range effort by Miroslav Stoch.
Immediately after the goal, however, Fenerbahce midfielder Colin Kazim-Richards was sent off, receiving a second yellow card for petulantly throwing the ball away after conceding a free kick.
Fenerbahce survived a second-half bombardment, Young Boys again hitting the woodwork, until conceding an 89th minute penalty, converted by the Costanzo.
Swiss champions Basel won 2-0 away to Hungary's Debrecen while Kazak champions Aktobe Lento pulled off the evening's shock result with a 1-0 win over Hapoel Tel Aviv thanks a second-half Samat Smakov penalty.
