Tommy Wright admitted Kilmarnock got what they deserved after a slow start to both halves at Fir Park saw them drop back into the Scottish Premiership bottom two.
It was the worst possible day for Killie, with Ross County and Hamilton both winning to leave the Ayrshire side two points above bottom side Accies.
Wright said: "We didn't start the game well, Motherwell looked sharper than us without really creating a lot.
"We worked our way into the first half but it's all right talking in the dressing room about going out and making sure we do the right things...
"It was a good finish but a poor goal to concede from us, too many mistakes in the build-up. Overall Motherwell deserved to win.
"We can't depend on other teams, I have been constant on that. We have to make sure we do our own thing and do it well, and we didn't do it well enough today."
Wright had Kyle Lafferty back in his team after the striker missed Monday's Scottish Cup shoot-out defeat by St Mirren through injury, but he could not take several chances after a quiet first half.
"We can't use Monday as an excuse in terms of the fact we looked lethargic at times," Wright said. "It was more a mindset.
"But the one thing was when we went down 1-0 there was a reaction, we tried to get back in the game.
"But it's criminal defending for the second goal and that was the game dead and buried. But it's always been the case we have to look after ourselves and we didn't do that."
Cole took his season's tally to 11 – 10 of them since Graham Alexander was appointed in January – and the Motherwell boss hopes to persuade the 25-year-old to extend his second spell at Fir Park beyond the summer.
Alexander, who was delighted with his side's commitment after they moved up to seventh, said: "We spoke to him quite a while ago about a contract and it's still ongoing. But we would be delighted to keep him. I think we can help him improve and get to the next level.
"I think he needs a consistency of being at one club for a certain amount of time. He has led a little bit of a nomadic existence up to now but he certainly has the qualities of having a top-level career and I believe we can help him with that, but we will see how it goes.
"He's been excellent. You can't ask any more of a number nine. His goals stand out for themselves, they are there for everyone to see, but his all-round contribution and link-up play and defensive side has been excellent.
"As it has been with the others, we can't win games with one centre-forward and a defender, it has to be a whole team performance."