The Fir Park side claimed their first home victory in over three months in the Clydesdale Bank Premier League with Lasley hitting the decisive strike eight minutes from time.
However, there was still time for one more chance for County but Hollis preserved the scoreline with a superb one-handed stop to deny Scott Boyd's header.
"I was on the line and thought Boyd had scored the equalising goal. I think at that point if it goes in you start to think it's going to be another one of those days," said Lasley.
"It was a great stop and he has done very well since he came in, everyone knows Darren Randolph is a top-quality keeper and Lee has filled his boots admirably.
"A save at one end can be as important as a goal at the other and that certainly was the case with Lee's save.
"Anything he was asked to do in the game he did well and the save at the end was probably the most important."
County took the lead after 17 minutes when Rocco Quinn unleashed a sensational 30-yard strike, but their advantage was short-lived as Michael Higdon headed home Chris Humphrey's pin-point delivery eight minutes later.
Nicky Law then capitalised on some slack possession to fire Motherwell ahead in the opening minute of the second half, but the visitors responded positively and Quinn hammered home at the second time of asking to level.
Both sides showed a determination to grab a winner and Lasley clinched the game for Motherwell when he flashed an effort beyond Mark Brown.