The Ibrox boss admitted on Thursday that his hopes of finally securing the extra midfielder he has long desired hung on whether his team could make it to the money-spinning group phase for the third year running.
Gerrard picked out Arfield out for special praise this week after an impressive return to form and the midfielder repaid his manager's show of faith with the opener seven minutes into the second half, with skipper Tavernier pouncing moments later for his seventh goal in as many games.
It is on nights like this that Ibrox has built its reputation for being one of Europe's most atmospheric venues, but even with the blue stands lying empty and a late goal from defender Marcao, Fatih Terim's men could not put a dent in Gerrard's incredible Europa League record.
Under the former Liverpool midfielder, Rangers have now racked up 19 qualifiers without losing a single leg and will now proudly take their place in Friday's draw in Nyon.
Gerrard named the same team that put four past Willem II last week as Ianis Hagi returned to face his father Gheorghe's old side.
The Turkish giants, meanwhile, started with Colombia ace Radamel Falcao and Gerrard's former Liverpool team-mate Ryan Babel.
The new handball rules have been causing havoc in England but Latvian referee Andris Treimani saw nothing untoward early on as the hosts put in a big appeal when a Connor Goldson header appeared to strike Martin Linnes' arm, which in fairness was down by his side.
It was a positive start from the Light Blues, with skipper Tavernier throwing over a delicious cross which just evaded Arfield's outstretched toe, while centre-back Christian Luyindama rescued the hosts as he headed a goal-bound Goldson header clear from under his own crossbar.
But Rangers were perhaps guilty of showing Galatasaray a bit too much respect, refusing to press the visitors with the usual ferocity they would normally muster when roared on by a full house.
A mix-up between Goldson and Arfield allowed Galatasaray to break at speed but there was a lucky let-off as Sofiane Feghouli's wild finish ended up in the Copland Rear.
Galatasaray had a penalty shout of their own on the half-hour mark as a Feghouli cross appeared to strike Arfield on the arm but again the award was not given.
Felip Helander made a vital interception to stop Feghouli bursting through on Allan McGregor's goal just after the break – and its value was underlined moments later as Arfield swept Rangers ahead.
Tavernier fired a ball into Hagi's feet just as Alfredo Morelos dragged his defender away with a clever run.
That opened up a huge gap in the Galatasaray defence into which Arfield burst as he collected the Romanian's pass before finishing with the outside of his right foot.
The small smattering of Rangers players and officials inside Ibrox burst into celebration.
The noise levels were not quite the same but the delight was clear to see.
And there was more to celebrate seven minutes later as Tavernier started and finished the move which led to the second. Receiving a pass from Goldson, the full-back drove up field before passing to Hagi.
The right-back carried on his run and found himself climbing above Linnes as Borna Barisic's cross spun up off Omar Elabdellaoiui's leg, to power a header into the top corner.
Galatasaray were at desperation point now. They screamed loudly for a penalty as Younes Belhanda went down under an Arfield challenge but again Treimani looked away.
Marcao angled a fantastic header past McGregor from Omer Bayram's corner with three minutes left but it was too little, too late for the visitors.