This result means that there are no Scarborough Sunday League teams left in the competition.
United switched their game to the 3G pitch at Malton Community Sports Centre, but could only raise 11 players for the fixture and this shortage of subs proved costly.
Gate opened the scoring after only five minutes and a misfiring United side were hit by a second goal on 15 minutes.
The visitors were then reduced to 10 men on the half-hour mark when one of their players was dismissed for a late challenge on Johnny Woodhead, but the hosts were also left a man short as Woodhead was unable to play any further part due to an ankle injury suffered in the challenge.
The away side pulled into a 3-0 lead after the interval, but the hosts grabbed a goal back when the impressive Jack South fired home.
United’s hopes of a fightback were ended shortly after when James Pilmer limped off with a stubbed toe and was unable to return to the action.
With Itis Itis pushing for a second goal Gate broke on the counter late on to seal their place in the quarter-finals, where they will play host to Layfield Armns on February 7.
United boss Aaron Hepworth said: “We were pretty poor today, I know we have not had a game for a while but neither had the opposition so we cannot use that as an excuse.
“Our man of the match was probably Jack South, while Chris Hannam was also on top form.”
Hepworth, who took over as boss from James Prethero just before Christmas, added: “We are on the look-out for new players to bolster our squad.
“Anyone wanting to join us can contact me on 07904161680 for more information, or come along to training on a Wednesday night at Pindar Leisure Centre at 6.30pm, which is also a training session for Westover Wasps’ two Saturday League teams.”