Kris Boyd volleyed Rangers into an early lead before Merouane Zemmama twice came close for the visitors.
Kris Boyd's superb turn and shot opened the scoring after 17 minutes and Kyle Lafferty volleyed Rangers' second.
The teenager had tested goalkeeper Lukasz Zaluska with a first-half drive and Kris Boyd curled an effort wide.
At the other end, Lee Mair made a crucial block to divert Kris Boyd's shot out for a corner.
Kris Boyd squandered a great chance to level when he headed over from close range but Rangers produced little else.
Cousin responded by netting the opener before Steven Naismith, Chris Burke and Kris Boyd helped themselves to a goal apiece.
Kris Boyd put Walter Smith's side ahead after 11 minutes with a neat finish from the edge of the box.
"That should give them a lift and I was very happy to see Kris Boyd given a start, " he said.
Kris Boyd hit the bar for the visitors with a first-half header and the Pars' Jim Hamilton flashed a header wide.
BBC: SPORT | Football | Scottish Cups | Dunfermline Ath 0-2 Rangers
Kenny Miller showed pace and poise to meet a Kris Boyd through ball and slot past Artur Boruc in eight minutes.
Steven Whittaker came into midfield in the absence of the suspended Kenny Miller, with Lafferty moving up front to partner Kris Boyd.
Birmingham forward McFadden had a frustrating night on the wing before a brief spell partnering substitute Kris Boyd up front late on.
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But the Teessiders looked most likely towards the end with Ruddy saving brilliantly from Barry Robson and Kris Boyd and Hoyte also going close.
Rangers looked to take advantage of a growing number of set plays but neither Kris Boyd or Edu could find the target with headers.
Kevin Thomson returned from injury to make his European debut for Rangers and Chris Burke was given the task of supplying ammunition for Novo and Kris Boyd.
BBC: SPORT | Football | Europe | Rangers 4-0 Hapoel (agg 5-2)
Kris Boyd fired the home side into a first-minute lead and grabbed a second from the penalty spot on 27 minutes after Nacho Novo had been fouled.
In the light of Rangers' premature exit, former Ibrox defender Craig Paterson believes Kris Boyd's failure to capitalise on a clear opportunity sealed the club's fate against Stuttgart.
With his side showing no sign of a breakthrough, Le Guen resorted to a double substitution, with Kris Boyd and Chris Burke replacing Dado Prso and Libor Sionko.
BBC: SPORT | Football | Scottish Premier | Dunfermline Ath 1-1 Rangers
Rangers team-mates Kris Boyd and Lee McCulloch have been recalled.
In 57 minutes Kris Boyd rose to flick on a long ball into the path of Kenny Miller, who passed to the advancing Davis to shoot past Graeme Smith.
Rangers manager Walter Smith laid down the gauntlet to Panathinaikos, abandoning his tried and trusted 4-5-1 formation and fielding strikers Kris Boyd and Novo in an attacking starting XI.
The Hibs manager steered clear of any controversy about the decision to award Rangers a penalty, when Sol Bamba challenged Kenny Miller, from which Kris Boyd put the home side 2-0 ahead.
With Kris Boyd a peripheral figure, thanks in part to the attentions of McLaughlin, he was substituted midway through the second half for Kyle Lafferty, rumoured to be attracting interest from Wolves.
He tried to point the finger at Kris Boyd for being a traitor to his country when the Rangers striker actually said it was the manager for whom he did not want to perform.
Jelavic - who scored 21 goals last season, including three against Celtic in the Europa League - is viewed as a replacement for Kris Boyd, who left Ibrox for Middlesbrough under freedom of contract.
It was nearly 2-0 after 16 minutes when a poor kick out by Lukasz Zaluska was headed forward by Pedro Mendes, only for Dixon's vital touch to divert diverted the ball wide with Kris Boyd lurking.
Their complaints came in the wake of Murray's revelation that one key player would have to be sold this month to balance the books, after the club accepted an offer from Birmingham for top goal-scorer Kris Boyd.
Top scorer Kris Boyd was restored to the front line after impressing as a midweek substitute while he aims to end an uncharacteristic six-game run without a goal, which would be his 100th in the league for Rangers.
Kris Boyd almost had a clear run at goal after four minutes when he latched onto a Velicka pass but Darren Barr made an excellent challenge on the edge of his own penalty area to halt the striker's progress.
Rangers began to threaten and a superb thrusting run from Nacho Novo, retaining his place after two goals on Tuesday, set up Kris Boyd, but his strike partner curled the ball wide of the post from the edge of the box.
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