Lazio and Steaua out of Europe
Elsewhere, Lazio's miserable season continued when they were dumped out after a 2-1 defeat at Salzburg while former European Cup winners Steaua Bucharest were also eliminated.
Salzburg, who have a 100 percent record from five group games, went through along with Benfica, Everton, Fenerbahce, Villarreal, Hapoel Tel Aviv and Hamburg SV, all with a match to spare.
They joined Werder Bremen, Galatasaray and holders Shakhtar Donetsk who had already qualified.
Group A leaders Ajax fell behind to a second-minute goal from Dorin Goga but levelled four minutes later through Uruguay striker Suarez, the Dutch league's top scorer with 17 goals.
Demy de Zeeuw's first-time ball freed Marko Pantelic to race through for the winner at the start of the second half.
Timisoara then had Eder Bonfim dismissed for hauling down Mitchell Donald in the box in the 50th minute, only for Suarez to see his penalty saved by Costel Pantilimon.
Pantilimon was sent off after giving away another penalty 12 minutes later for fouling Suarez as he raced clear on goal. The Uruguayan then produced an almost replica effort which was saved by replacement keeper Pedro Taborda.
Timisoara, bottom with two points, were eliminated.
In Group G, Cameroon's Somen Tchoyi scored a superb winner for Austrians Salzburg against Lazio, cutting in from the right and beating his marker twice before chipping nonchalantly into the net.
Lazio dropped three points behind second-placed Villarreal, 2-0 winners at Levski Sofia with goals from Giuseppi Rossi and Marcos Senna, and cannot catch the Spaniards who have a better head-to-head record.
Steaua were left bottom of Group H with three points after a 1-1 draw at Moldovan champions Sheriff Tiraspol, a late equaliser from Colombian Juan Toja cancelling out Senegal forward Amath Diedhiou's goal for Sheriff.
In the same group Uruguay captain Diego Lugano headed the only goal to give leaders Fenerbahce a 1-0 victory at former England manager Steve McClaren's Twente Enschede, a result which sent the Turks through.
Argentine forward Javier Saviola scored one and set up a second for Fabio Coentrao as Group I leaders Benfica won 2-1 at BATE Borisov to qualify.
They were joined by English Premier League strugglers Everton, who won 1-0 at AEK Athens with an early goal from Diniyar Bilyaletdinov.
Marcell Jansen and Marcus Berg scored early in the second half to give Hamburg SV a 2-0 win at home to Rapid Vienna which took the Germans top of Group C with 10 points and guaranteed them a place in the next round.
Hamburg's victory also allowed Hapoel to go through despite losing 2-0 at Celtic.