Llanelli, who sacked their director of rugby Phil Davies earlier in the week, roared into a 20-5 half-time lead with Munster leaking two soft tries to Dafydd James and Matthew Watkins.
But the visitors were able to turn the tables in the second period.
Winger Ian Dowling cut through for his second try of the game and replacement out-half Paul Warwick's late heroics - he scored 11 points including a monster 90th-minute penalty goal - saw the men in navy nick it at the death.
Quinlan said: "We never gave up. We were disappointed at half-time and the score-line could have been even worse.
"We made far too many mistakes and missed tackles and Llanelli punished us. We were comfortable with our positions but we didn't make enough one-on-one tackles.
"At the break, we tried to remain positive and get an early score in the second half to get us back into the match. Thankfully we did."
Given the sides' current status with the Scarlets in poor form and without a coach and Quinlan and company preparing for a Heineken Cup final, Munster could have been accused of taking the Welsh region lightly but the Tipperary man insisted that was not the case.
"Llanelli played really well in the first 40 minutes and we didn't take them lightly.
"It was great character from us in the end that won it. We're delighted at the way we dug in and got the result in injury-time."
But missed tackles by Munster players in the lead-up to the tries scored by James and Watkins had the province's defence coach Tony McGahan concerned.
"It wasn't a great display and I was tearing my hair out when we conceded that second try (to Watkins).
"But you can't question the character or commitment of these players. It was fantastic. It's a goal for us to finish the season as strongly as we can," he said.
McGahan also singled out Warwick, who landed that match-winning kick, for special praise.
"It was a great kick by Paul to win the match. We weren't convinced that he would score from that range - just inside his own half - but he struck it brilliantly."