Ole Gunnar Solskjaer's side were well below their best in a scrappy fifth round tie at freezing, snow-dusted Old Trafford.
But a fine finish from Scotland midfielder McTominay ensured United advanced to the last eight for a seventh successive season.
United last won the FA Cup in 2016 and McTominay's third goal in three games kept them in the hunt to lift the trophy for the 13th time.
After the frustration of the late Everton equaliser that dented their Premier League title challenge on Saturday, this was an essential result to lift the mood at United.