Vine scored the only goal when he drove home from close range after Murray Davidson had bundled the ball towards the line when Mehdi Abeid's corner bounced in the six-yard box.
"I thought the first half we handled them very well," Lennon said.
"The pitch certainly wasn't conducive to the way we like to play but we caused one or two problems by going a bit more direct.
"We handled them very well with their threat from set plays, which they are very good at.
"What was very disappointing in the second half was the manner we conceded the goal. That's five games in a row now that we haven't lost a goal from open play.
"And for me it was the worst ball that came in from their set plays, and we didn't handle it. That's the difference between walking away with one point and no points."
Lennon's side are now eight points off the top six and their home tie with Hearts on Wednesday looks like a last chance for both clubs to have any hope of getting there before the split.
But the Paisley side could still be missing winger Gary Teale, who has an abductor strain.
Lennon said: "I certainly know what our downfall is at this minute - that's three games in a row that we haven't scored. I know where the problem lies - the quality coming in from wider areas hasn't been the same since Tealey went out."