Kick-off had been delayed by 15 minutes as Swindon were not expecting the crowd of 2,327 that turned up.
Andy Carroll scored the vital goal with a headed winner four minutes from the end of extra time to beat 10-man Swindon in the semi-finals.
United Make It To Cup Semi-Finals
United led after 28 minutes when Carroll met Doninger corner at the far post and headed across goal for Ryan Donaldson to force the ball over the goal-line with his head.
But Swindon were level on 69 minutes when centre-forward Lukas Jutkiewicz struck a 25-yard drive low past Mark Cook into the left corner of the net and the tie went into extra time.
Two minutes into the extra period Swindon skipper Chris Allen picked up a second yellow card for fouling LuaLua and United finally got the break through in the 116th minute. Carroll scored from another Doninger corner.
Newcastle will play either Sheffield United or Cup-holders Liverpool, who meet at Bramall Lane on Thursday.
The first leg of the semi-final will be at St James's Park.
Swindon Town: Stewart, Westall, Kennedy, Doe, Evans, Hammond, Morris, Allen, Jutkiewicz, Hyde, Hopper. Subs: Willmoth, Macklin, Thompson, Jamie Taylor, Roberts.
Newcastle United: Cook, Cowan, Bertram, Doninger, Morris, Francis, Godsmark (sub Danquah 110), Cavener, Carroll, Donaldson, LuaLua. Subs: Critchlow, Marwood, James Taylor, Debbage.