The 18-year-old was thrust into the United starting XI for Thursday's Europa League last-32 second-leg clash with Midtjylland when Anthony Martial pulled up injured in the warm-up.
Two goals in that 5-1 win helped United into the next round and Rashford showed no signs of slowing down against the title-chasing Gunners.
He scored twice in the space of three first-half minutes – netting his first Premier League brace at exactly the same age as club captain Wayne Rooney when he registered his (18 years, 120 days).
United alumnus Danny Welbeck pulled one back for Arsenal before the interval but Ander Herrera made it three in the 65th minute – Rashford providing the assist.
Ozil once again reduced the deficit four minutes later, but Arsenal left empty-handed and sit five points behind leaders Leicester City, while United climb back to fifth.
Welbeck, who was deemed surplus to requirements by Louis van Gaal at United, came into the Arsenal starting line-up, while the Dutch manager – now desperately short of attacking options due to an injury crisis - was once again forced to deploy Rashford.
The first chance of the match came courtesy of an unlikely source, though - Nacho Monreal firing straight at the returning David de Gea after being played in by Ozil.
Rashford taunted the Arsenal defence with an early surge into the box, drawing the foul – and a free-kick on the edge of the area – from Gabriel Paulista, but Memphis Depay could only fire the set-piece straight at goalkeeper Petr Cech.
United had conceded three goals by the 19th minute in the reverse fixture but Arsenal, who positioned Welbeck out wide with Theo Walcott down the middle, did little to suggest they could carve open the home defence with the same lethal touch this time around.
And Rashford's dream week continued when Arsenal failed to clear a Guillermo Varela cross from the right, with the young striker placing a shot beyond Cech from the centre of the box.
The home fans had barely caught their breath when, three minutes later, Rashford headed in a second from a Jesse Lingard delivery – a goal with roots firmly in the United academy.
Rashford is a graduate of Fletcher Moss Rangers, for whom Arsenal's Welbeck also once played, and the community football club celebrated a treble when the latter headed beyond De Gea from an Ozil free-kick to make it 2-1 at the break.
United continued to impress after half-time and, following blocked shots from Morgan Schneiderlin and Depay, Rashford teed up Herrera to score from 20 yards, although the effort owed more than a little to a huge deflection off Laurent Koscielny.
Ozil's volley brought Arsenal back to 3-2 when the ball bounced kindly for him some 12 yards out in the 69th minute, before tempers briefly flared between Aaron Ramsey and Herrera.
Van Gaal showed his displeasure at a decision by referee Craig Pawson by dramatically throwing himself to the ground on the touchline late on, but it is Arsenal who must now pick themselves up after a string of disappointing results in the FA Cup, Champions League and now the Premier League.
