Saints turned the game around after Gary Caldwell had given the hosts the lead, with Lambert and Morgan Schneiderlin on target.
But the good work was undone when Shaun Maloney levelled in the dying seconds at the DW Stadium.
Lambert said: "I'm frustrated about conceding in the last minute, it was a very bad goal to give away.
"Obviously when the dust settles and you have time to think about it then we're going to think of it being a good point. But right now it doesn't feel like that, it's a bitter pill to swallow. We should have had three points.
"I thought the performance was brilliant. Throughout the 90 minutes we pressed them, we harried them and we put them under a lot of pressure, and they're a really good football team.
"Last time we played them we got taught a lesson but I thought this time around we showed them how much we've improved and we deserved the three points.
"We've been good of late at not conceding many goals. The last two games we've conceded two and these were poor goals, both of them. We need to get back on the training pitch and do better next week."