Fenerbahce were eliminated after a stunning 1-0 home defeat by Young Boys, while Celtic went out despite a 2-1 win over Braga and Dinamo Zagreb were stunned by Moldovan champions Sheriff Tiraspol, losing their tie on penalties.
Ajax, held 1-1 at home by Greek side PAOK last week, went through on away goals despite squandering a 3-1 lead in the third qualifying round, second leg tie.
Portuguese forward Vieirinha gave PAOK a half-time lead before Suarez, last season's Dutch League top scorer with 35 goals, and Siem de Jong scored twice within three minutes for the Dutch team just after the break.
Rasmus Lindgren seemed to make the tie safe for the former European champions with a third in the 55th minute.
But Ajax, who last reached the group stage in 2005/06, had to cling on after Dimitris Salpingidis pulled one back within a minute and Vladimir Ivic scored a stoppage-time equaliser.
Suarez scored three goals at the World Cup to help Uruguay reach the semi-finals. He was also involved in controversy in the quarter-final against Ghana, when his handball stopped a goal-bound shot on the line and he was sent off in the last minute.
Fenerbahce, quarter-finalists in 2008, were clear favourites against Swiss league runners-up Young Boys, who have never played in the group stage, after a 2-2 away draw last week.
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Instead, the visitors took the initiative and Cameroon striker Henri Bienvenu took one of many chances when he scored from a narrow angle in the 40th minute.
The Turks, missing Uruguay captain Diego Lugano with a knee injury, had Miroslav Stoch sent off in the 53rd minute when he was shown a second yellow card for diving.
Braga stayed on course for their first group stage appearance after Paulo Cesar gave them a 20th-minute lead at Celtic. Although Gary Hooper and Efrain Juarez replied for the Scots in the second half, the Portuguese side ran out 4-2 aggregate winners.
Russia's Zenit St Petersburg qualified against Romania's Unirea Urziceni - group stage participants last season - when Portuguese striker Danny scored the only goal of the tie.
Former AC Milan and Chelsea striker Andriy Shevchenko set up two goals to help Dynamo Kiev win 3-1 at Ghent and complete a 6-1 aggregate win.
Ajax, Young Boys, Braga, Dynamo and Zenit are all in the non-champions half of the draw and will be joined in the next round by Tottenham Hotspur, Sampdoria, Werder Bremen, Auxerre and Sevilla.
The winners qualify for the group stage.
In the champions half, Austria's Salzburg demolished Omonia Nicosia of Cyprus 4-1 to win 5-2 on aggregate.
Brazilian striker Cleo scored at the start and the end of the match go help Serbian champions Partizan Belgrade win 2-1 at HJK for a 5-1 aggregate
