Bryan Redpath’s side only needed a point to take second place but they were also seeking some much-needed form to take into the play-offs.
Having returned to winning ways against Richmond after suffering a five-match losing run, Carnegie backed up with this victory.
Winger Jonah Holmes also crossed in the 53rd minute and Mike Mayhew claimed the visitors’ final try in stoppage-time, Joe Ford contributing 18 points via four penalties and three conversions.
On securing second spot, No 8 Burrows admitted: “It’s massive. It’s been a tough six seven weeks on the back of five losses but we got what the squad deserved. We targeted a top-two finish in pre-season and if we’d have not got that it would have been very disappointing.
“It would probably have marred a very good season so to secure that we’re very happy.”
Burrows, who joins Premiership Newcastle Falcons at the end of the season, conceded Carnegie were still far from their best.
“I’ll own up and think we were very lethargic as a team,” he said.
“We were very poor in warm-up and it probably took us 40 to 50 minutes to get into this match.
“We were very happy to go in at half-time leading, having played so poorly and in fits and starts.
“As a forward pack, we were very disappointed with how we performed – it’s the poorest we’ve been all season in terms of our line-out – but we stayed positive and we’ll get better from it.”
On Saturday, Carnegie host third-placed Ealing Trailfinders, whom they are also set to face in the semi-finals although Doncaster Knights, who welcome leaders London Irish and are certain of finishing at least in fourth position, could yet usurp them.
“We’ll come in on Monday and start building towards Ealing,” added Burrows.
“If we end up playing them in the semis, too, so be it. It will help on analysis as we’ll know all about them and I don’t think they’re going to change their style much in the play-offs.”
Rotherham Titans, meanwhile, suffered an embarrassing 82-3 defeat at Irish in their final game of the season.
Coupled with Richmond’s shock win over Cornish Pirates, it means the South Yorkshire side fell bottom, rounding off a dismal campaign and illustrating the size of the task facing Andy Key as he takes over at Clifton Lane for 2017-18.