The Gers were in a commanding position at the interval thanks to a brace from the 26-year-old, before their lead was embellished by a fine Kieran Dowell header shortly into the second period.
Possession of the match ball was secured for the North Macedonian on 73 minutes when he converted from the spot following a foul on Rabbi Matondo, yet perhaps the finest strike of the evening came from Thelo Aasgaard when his 30-yard free-kick nestled into the top corner with four minutes to play.
It was also an evening where Tochi Chukwuani stepped onto the Ibrox pitch for the first time and 17-year-old Zebedee Lawson made his senior debut for the club as the home side chalked up their sixth victory on the spin.
The famous old stadium was abuzz with excitement as the teams walked out following the unveiling of new recruit Andreas Skov Olsen, introduced for the first time as a Gers player alongside Chukwuani, among the substitutes this evening, and Tuur Rommens.
It was much changed Gers side from last weekend’s victory over Aberdeen at Pittodrie, with Liam Kelly captaining the side and one of nine changes from the win in the north-east five days ago.
Rangers last squared off against Annan back in 2022 in Scottish Cup duty, a year which yielded their most recent triumph in the competition, and the unique challenge that lower league opposition pose is never easy to navigate.
Indeed, it was the away side who had the first shot on goal in the contest when a bouncing ball broke to Tommy Muir from 30 yards out and the midfielder lashed over the crossbar on the half volley.
But with Rangers’ first foray forward they made a difficult task for the League Two side even harder.
The hosts were swift in their one-touch passing through the heart of the Annan midfield as Djeidi Gassama combined with Danilo to send Kieran Dowell beyond the opposing backline.
The midfielder couldn’t quite sort out his feet, but the ball broke invitingly for Miovski who took a deft touch inside and stroked a delicious finish beyond Charlie Albinson.
Rangers were certainly showing their ruthless side in the opening period as they doubled their advantage with their next chance of genuine note.
Again, their speed of play was too much to handle for the visitors as Gassama spun away from Wullie Gibson and found Kieran Dowell. He swiftly found Findlay Curtis in space to his right who stood up a delightful delivery for Miovski to nod home his second of the evening.
The crossbar came to Annan’s rescue six minutes before the interval, which would surely have put the game beyond doubt in the first half, when Clinton Nsiala lofted an unerring pass over the top of the visiting defence where Curtis beat the onrushing Annan goalkeeper and lofted his effort against the woodwork.
Miovski appeared a man on a mission to secure his hat-trick, which very nearly arrived moments later when a short corner routine unlocked a dangerous Gassama delivery where the North Macedonian international popped up at the far post and was denied from close-range by Albinson.
The second period began with the Gers once again in the ascendancy, extending their lead further inside three minutes.
It was another well-worked abbreviated set-piece routine as the hosts worked the space for Joe Rothwell to deliver; his cross met by Dowell who glanced a gorgeous header beyond Albinson who was left flat-footed.
Rangers were in the mood for more as Danilo curled over following smart play from Jayden Meghoma, before Curtis lashed wide when he let fly across the Annan goalkeeper on the right-hand side.
There was a special moment just nine minutes into the second half as Danny Röhl introduced Lawson to the fray, making his first senior appearance after impressing for the Under-19s this season.
Rabbi Matondo also sprung from the bench and shot off target after skipping dangerously inside in typical fashion off the left-hand side.
As the clock ticked beyond the hour-mark it was time for Chukwuani to make his Ibrox bow as he replaced Rothwell to a rapturous reception inside the stadium.
Despite having to wait a long time, Miovski’s third of the evening arrived with 17 minutes remaining.
Matondo was the architect as he was clipped inside the penalty area by Scott Hooper, leaving the referee in no doubt about the awarding of a spot-kick.
The forward grabbed the ball and strode forward from 12 yards to send the goalkeeper the wrong way.
They always say that you should save the best until last, and that was true with just four minutes remaining on the clock as Aasgaard tried his luck from a free-kick 30 yards out and he watched the ball sail into the top corner to put the icing on the cake for the Gers.
RANGERS: Kelly, Aarons, Djiga, Nsiala, Meghoma (Lawson, 54), Rothwell (Chukwuani, 62), Dowell (Aasgaard, 62), Danilo (Moore, 70), Gassama (Matondo, 54), Curtis, Miovski
Subs not used: Wright, Souttar, Diomande, Chermiti
ANNAN ATHLETIC: Albinson, Hooper, Kilsby (Gaffney, 79), Dixon (Goss, 57), Muir, Watson, P. Smith (McGowan, 70), A. Smith, Muir, Maxwell (Scott, 79), Gibson
Subs not used: Clarke, Deveney, Craik, Mumbongo, Carrick