Motherwell started the game brightly and almost took the lead in the sixth minute when Steven Hammell flashed a low cross across the face of goal, but John Sutton couldn't apply a strong enough touch to divert it goalwards.
However, Motherwell came back strongly and Lionel Ainsworth almost put his side ahead after 25 minutes, only for Dean Brill to make a sharp reflex save to deny him, but a minute later the winger scored the opener.
The 26-year-old burst away from the opposition defence before drilling a low shot beyond the reach of Brill.
In the 34th minute, Sutton made it 2-0 with a simple header from a teasing Hammell cross.
The visitors did threaten 20 minutes from time when substitute Ryan Christie dragged a long-range shot just wide of the far post, but Motherwell held on to secure a comfortable victory which ensured they kept pace with League Cup winners Aberdeen.