The north-east clubs made the drop after last-day defeats, ensuring Hull City and Sunderland survive in the top tier of English football.
Newcastle needed to better Hull's result, but while the Tigers lost 1-0 at home to an unfamiliar-looking Manchester United side the Toon also lost 1-0 at Aston Villa.
Newcastle's misery was complete when the centre-back David Edgar was sent off after a second booking in the 94th minute.
Middlesbrough were beaten 2-1 at West Ham. West Bromwich Albion had already been relegated.