The former Leicester City striker, who was given half an hour of action, netted in injury time to add a little bit of gloss to what was a very comfortable if not emphatic win against the side that was demoted from the senior game last May.
Andy Halliday, with a penalty, and Josh Windass were the other Rangers scorers as the Ibrox men made it three wins in a row without concession and will officially take their place in the knockout phase if they beat Stranraer on Monday night.
Joey Barton will no doubt feature in that Ibrox match having been left out last night due to the synthetic surface at Ochilview.
There was never an inkling of being in any difficulty as Mark Warburton gave vital game time to the majority of his new recruits although Windass had to go off at half-time with what looked like a thigh injury.
Clint Hill got 90 minutes beside Danny Wilson while Niko Krancjar got another 45 minutes as Windass’ replacement.
It was in stark contrast to the last visit to Ochilview in April 2013 when the post-liquidation Ally McCoist side to work extremely hard for a 4-2 win in the Third Division.
Rangers zipped the ball across the synthetic surface with pace and precision and The Shire players were unable to stop them.
It was Andy Halliday who got the ball rolling when he scored from the spot in 10 minutes.
Barrie McKay had been crudely brought down by Ross Fisher and Halliday made a neat job of the penalty driving a left foot shot into the right corner.
Perhaps the only complaint was that it took Rangers a further 25 minutes before they found a second goal.
Lee Wallace had forced Darren Dolan into a good save at his right post in 13 minutes while Fisher did brilliantly to clear a dangerous Josh Windass cross in 20 minutes as Kenny Miller waited for a tap-in.
Then in 27 minutes Dolan came to the Shire’s rescue again making an excellent save from Miller’s header.
However, the keeper was completely exposed in 35 minutes as Rangers produced a sparkling move.
Halliday sprayed a great pass wide to McKay who produced an accurate cross for Windass and the Englishman took one touch before stabbing a right foot shot past the stranded Dolan.
Miller missed a glorious chance three minutes later when he was picked out expertly by Wallace and skewed a right a right foot shot high over the bar from only eight yards out.
Dodoo was given his chance on the hour mark, replacing Michael O’Halloran, and eased his way in on his debut.
With Krancjar on for Windass, the Rangers supporters were entitled to think there were more goals to come but East Stirlingshire were resolute in their defending.
The chances were few and far between with Harry Forrester hitting the side netting in 57 minutes after some great work by McKay then his replacement, Jordan Thompson, forcing an excellent block out of Dolan in 76 minutes.
Then with 10 minutes remaining Krancjar flashed a right foot shot wide after Lee Hodson’s initial effort had been blocked.
Finally, in added time Rangers produced a third goal and it was a great moment for the latest recruit.
Lee Wallace ran onto McKay’s pass and cut the ball back for Dodoo who slid in to knock the ball home from close range.