Whittaker did well to block a shot from Tod for a corner as the home side rallied.
From the resulting corner, the ball fell again to the former Pars stopper who drove wide from just inside the penalty area.
In the 40th minute Brown pulled off a remarkable save from Miller's perfectly executed over-head kick.
Boyd drove wide from the same distance as the visitors finished the half strongly.
Rangers looked in no mood to offer any succour when the game restarted.
Brown again had to be at his best to smother an angled shot from Beasley.
But in the 54th minute Miller doubled the visitors' lead when Loons defender McNally hoofed his attempted clearance straight into the striker's path.
Miller flicked the ball over a defender's leg before lashing it high past Brown from 16 yards.
To make matters worse for the home side Smith was shown a straight red card with 20 minutes to go for a dreadful challenge on Beasley.
In the 84th minute substitute Kyle Lafferty threaded a lovely pass through for Niguez to bound onto and finish with aplomb.
It was Lafferty again who set up Miller in the dying moments for the striker to head home and confirm Rangers' passage through to meet Hamilton in the next round.
Forfar Athletic manager Dick Campbell:
"The finishes were high-class (by Rangers), you would expect that.
"But if you defend like that an amateur team will beat you.
"Not to hang young Macca (Stephen McNally) out to dry - he's been a very good player for us this year - but he's apportioned the blame for three of the goals and you just can't do that against that level of opposition.
"I'm not too unhappy but I just thought they didn't deserve to get beat by four."
Rangers assistant manager Ally McCoist:
"I was a little bit disappointed in the first-half. We got off to a good start with the early goal, but we overplayed at times.
"Sometimes we were trying to be too intricate with our passing.
"A long ball isn't necessarily a bad ball and I wanted some long passes to turn the Forfar defence.
"I felt that in the second-half we did that a lot more. But I'm delighted with the result and the clean sheet.
"We're in the hat again, so that's great."
Forfar Athletic: Brown, McNally, Tod, Tulloch, Elliot Smith, Gordon (Donachie 87), Winter (Fotheringham 74), Brady (McGuigan 74), Graham Gibson, Lilley, Campbell. Subs Not Used: John Gibson, Sean Kilgannon.
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