The North London club now stand on the brink of a 17th straight qualification for the lucrative pool stages after a dreadful start to their English Premier League campaign with defeat to Aston Villa.
A potential tricky trip to Istanbul instead ended up as a smooth ride as Ramsey, Kieran Gibbs and Oliver Giroud all got on the scoresheet.
The breakthrough after a dull opening half arrived in the 51st minute when Ramsey played in Theo Walcott down the right wing and the winger’s cross was allowed to reach Gibbs who crashed the ball into the roof of the net.
Ramsey then scored a fortunate second goal with 24 minutes left when Fenerbahce ‘keeper Volkan Demirel allowed a routine shot from outside the box to slip under his hands.
Giroud then planted a confident penalty in the top left hand corner after Michal Kadlec brought down Theo Walcott in the 77th minute.
The Gunners have a proud record of having qualified for the group stages of every Champions League competition since 1998, but they laboured during a lacklustre first half that featued no goals and few chances.
Fenerbahce meanwhile were looking to reach the pool stages despite a potential ban looming over them for match fixing that has gone to appeal and will be decided on the eve of the draw.
The odds were stacked against them having beaten English opposition only once in European Cup history dating back to 1969 when they eliminated Manchester City 2-1 on aggregate.
The Istanbul club rarely threatened the Arsenal goal throughout the match and Arsene Wenger’s team were able to take a firm stranglehold on the tie with three unanswered second half goals.
Elsewhere Wednesday, there were 1-1 stalemates at the encounters between Steaua Bucharest and Legia Warsaw, and Schalke and PAOK.
Swiss side Basel, beaten by Chelsea in the UEFA Europa League semi-finals last season, saw off Ludgorets Razgrad 4-2 while Austria Vienna defeated Dinamo Zagreb 2-0.