Ameobi netted the first in the ninth minute, capitalising on an error by the home defence before slotting past Andy Gruenebaum.
Gruenebaum was dismissed for a handball outside the area on the stroke of half-time and Newcastle took advantage after the break, claiming further goals through Fabricio Coloccini and Haris Vuckic.
Walk in the park: United celebrate following Ameobi's opener
Gruenebaum will still be eligible for the Major League Soccer match at Real Salt Lake on Saturday.
Crew coach Robert Warzycha thought the straight red was harsh on his keeper.
'You have fans in the stadium who came to see a good game and came to see the players,' he said. 'It was difficult for us to play the second half and for the fans to enjoy the game.
'It was the referee's decision, but if he had given a yellow card, everyone would have appreciated that.'
Flashpoint: Gruenebaum handles outside the box and is shown a straight red
Newcastle manager Alan Pardew agreed.
'The referee, probably knowing that he'd be criticized heavily if he didn't send him off fully, thought that was the correct decision,' he said.
'I'd prefer he hadn't. Of course we didn't want that for Crew. Certainly the second half was a little bit easier for us.'
Getting shirty: Fans scramble to grab a player's jersey thrown into the crowd
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