The 18-year-old winger, who had previously made such a positive impression during his ten substitute outings under Steven Gerrard this season, marked his first start for the Ibrox club with two goals in a comfortable victory over second tier Ayr United.
Nikola Katic and Alfredo Morelos were the other scorers as Rangers maintained the impetus they have enjoyed under Gerrard so far. The only sour note for the manager was a foolish yellow card incurred by Morelos which earns him a suspension for next month’s semi-final.
Before first half goals from Katic and Middleton put Rangers firmly in control of the tie, Ayr were at least true to manager Ian McCall’s pre-match promise that they would attempt to take the game to their hosts.
The free-scoring Championship leaders were the first to threaten, providing an early test for Wes Foderingham. The goalkeeper, making only his second appearance of a season in which he has lost his number one status to Allan McGregor, reacted sharply in the 10th minute to touch over a rising shot from Robbie Crawford after Rangers had struggled to clear an Ayr corner.
The visitors caused further anxiety for the Rangers defence three minutes later when Alan Forrest showed fine footwork to beat Katic on the left before whipping in a low cross which Andy Halliday did well to cut out.
Showing six changes to the starting line-up from Sunday’s thumping league win over St Johnstone, Gerrard’s side took time to find their rhythm but the momentum swung firmly in their favour when they made the breakthrough in the 16th minute.
Middleton’s pace and purpose down the left allowed him to fire the ball low across the six yard box. Alfredo Morelos was unable to connect and Ayr defender Daniel Harvie turned the ball behind for a corner. Ryan Kent swung it over from the right and Katic rose highest to power a header beyond Ayr’s on-loan Celtic ‘keeper Ross Doohan.
It was the big Croatian defender’s second goal in Rangers’ colours and he almost added to his tally eight minutes later when, after his run wasn’t picked up by the Ayr defender, he came agonisingly close to getting on the end of a Middleton free-kick.
After the earlier adventure they had shown, Ayr were finding it more difficult to make any progress towards Foderingham’s goal but they had a decent penalty claim waved away by referee Bobby Madden when Forrest went down under Jon Flanagan’s untidy challenge.
Rangers doubled their lead in the 31st minute as Middleton latched onto a cleverly lofted through ball from Halliday and raced into the penalty area to drive a firm left foot shot beyond Doohan.
With Kyle Lafferty cup-tied, the onus was firmly on Morelos to lead the line for Rangers in this competition but the man who scored a hat-trick in the previous round against Kilmarnock blotted his copybook with another needless and costly bout of indiscipline eight minutes before half-time.
The Colombian striker was floored by a crunching challenge from Ayr captain Jamie Adams and, after play had been allowed to continue, reacted petulantly to referee Madden’s handling of the situation. Halliday, captain for the night in the absence of the rested James Tavernier, rushed across to try and calm Morelos but was too late to prevent him collecting the booking - his second in the tournament - which rules him out of the semi-final.
Morelos was allowed to make some amends when he made it 3-0 four minutes into the second half, stabbing the ball home from close range after a Kent corner had been scuffed into his path by Ross McCrorie. He was substituted immediately after what was his eighth goal of the season, replaced by Eros Grezda.
With the tie effectively wrapped up, Gerrard also introduced fit-again Jordan Rossiter and sent on teenager Stephen Kelly for his debut. The tempo of the contest had dropped but Rangers burst into life again to make it 4-0 in the 70th minute when Middleton grabbed his second of the night. Halliday was the provider once more with a piercing through ball for the youngster who guided a low shot beyond Doohan from around 16 yards.
Middleton missed a good chance to claim a hat-trick, shanking a shot wide in the closing stages, but nothing could take the shine off a special night for the richly promising teenager.
Rangers: Foderingham, Flanagan, Worrall, Katic, Halliday; Dorrans (Rossiter 56), Ross McCrorie (Kelly 69), Ejaria; Kent, Morelos (Grezda 50), Middleton. Subs not used: Robby McCrorie, Wallace, Goldson, Arfield.
Ayr United: Doohan, Smith, Adams, Rose, Harvie; Geggan (Moffat 59), Murdoch (McDaid 65), Kerr, Crawford; Forrest; Shankland (Moore 76). Subs not used: Hare-Reid, Ferguson, Bell, Docherty.