The Scottish Championship club made it seven wins from seven matches with a five-star cruise at the Excelsior Stadium which saw Nicky Clark, Andy Halliday and Martyn Waghorn on the scoresheet in the first half before Dean Shiels and James Tavernier added late goals.
The 18-year-old German-born USA youth international shone in an attacking midfield role, having a hand in Halliday's 14th-minute strike, before playing a luscious through ball for Waghorn to make it 2-0.
Zelalem later then set up Tavernier in the 88th minute as Warburton's side rounded out the triumph in emphatic fashion.
Ball, in his full debut, started at centre-back and played a full 90 minutes while Oduwa also went the distance with a pleasing performance in an attacking three.
Both Ball and Oduwa were clearly delighted with the devastating win:
Good 5-0 win in the cup, another clean sheet and onto the next round, fans superb as usual! @RangersFC
— DominicBall (@DominicBall6) August 26, 2015
Good 5-0 win @RangersFC tonight, fans were amazing again they made it feel like a home game tonight! #WATP#RFCpic.twitter.com/UyhkaUIvyU
— Nathan Oduwa (@nathanoduwa) August 26, 2015