The Magpies were on target after eight minutes against last season's second-placed side in Scotland, with Haris Vuckic reacting quickest to tap home from close range.
Yoan Gouffran doubled Newcastle's advantage soon after, with a neat finish just inside the penalty area.
Motherwell were able to reduce the deficit on 26 minutes in style through Iain Vigurs, who smashed past Rob Elliot's neat post from 20 yards.
Early in the second half Motherwell got back in the match from the penalty spot, with James McFadden calmly chipping his spot kick down the middle to deceive Elliot.
On 64 minutes, Newcastle got their fourth and final goal to clinch a credible victory as Sylvain Marveaux slammed home from eight yards.