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 Sutton couldn't apply a strong enough touch to divert it goalwards.
McKay tested Nielsen once more seven minutes later, but the Faroese goalkeeper denied him again.
However, Motherwell came back strongly and 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.