Bath actually outscored their hosts by two tries to one in Belfast but those scores came from Ulster mistakes as the 1998 European Champions failed to threaten the Ulster line as regularly as Meehan would have liked.
"We were fortunate to be up but then we should have taken advantage of that, there's no doubt about it," said Meehan after Bath had raced into a 15-3 lead after just 22 minutes.
"Our intensity in attack has to be the same as it is in defence because defensively it was a marvellous effort.
"I don't have the stats for possession and field position but it would be largely in Ulster's favour.
"If we can just get our share of the ball in an attacking area and match that defensive intensity, then we're going to be in with a shout. But until we do that we're going to give good teams, teams like Ulster, opportunities."
Bath may have lost out to the Northern Irish province for the third successive time in 14 months but their Heineken Cup dream remains very much alive after Airoini's surprise win over Biarritz.
Last season's beaten finalists became Aironi's first victims since their inception this summer, leaving qualification for the quarter finals still very much up in the air at the halfway mark of Europe's premier club competition.
Bath currently sit in third place in the pool following one win and two defeats but they are just four points behind Biarritz and two behind Ulster.
Successive home wins over Stephen Ferris and co this coming Saturday and Aironi on the second weekend in January would leave a last-eight place very much within their grasp come the trip to Biarritz in Round 6.
"This group is still anyone's ball game," added Meehan.
"It's 11, nine and seven in points, so it's anyone's ball game with three matches to play.
"We need to work on two or three key areas this week and then, obviously, turn around the result against Ulster.
"We've got two of the next three games at home and we've got to Biarritz which we always knew would be a big game - it's shaping up that way."