Sam Winnall had put Barnsley ahead after five minutes and Marley Watkins restored their lead 73 seconds after Lansbury equalised, but after Watkins was sent off in the second half for an attempted stamp on Thomas Lam the hosts capitulated.
“It’s a perfect reaction because I wasn’t happy with the first-half performance, especially the first 20 minutes – we lost all the duels and didn’t have control of the ball,” said Montanier.
“The target was to keep a clean sheet, but sometimes the team want to play with their character. It’s better to win 5-2 than draw 0-0.
“After the victory at Ipswich we needed another victory because we’re not well ranked and before this game we were one point from the relegation zone.
“To score five goals is perfect for the fans.”
Barnsley went in front when Winnall headed home after five minutes following good work down the right by James Bree.
Lansbury equalised from distance after a one-two with Pajtim Kasami from a free-kick 30 yards out, but Watkins finished well from Conor Hourihane’s clever chip over the top moments later.
Vellios levelled the scores again with a miscued effort before Lansbury put the visitors ahead with a fine header during a frantic opening 45 minutes.
Osborn grabbed a fourth for the away side after half-time with a well-placed header shortly before Watkins saw red.
Lansbury completed his hat-trick from the penalty spot late on after substitute Matty Cash was brought down in the area.
“I enjoyed his three goals – it’s his first hat-trick, I hope it’s not the last,” Montanier said of the 26-year-old.
“He has the technical quality. Former Arsenal players are very good quality.”
Barnsley manager Paul Heckingbottom was left to rue a series of defensive mistakes from his side, who had kept clean sheets in their last two outings but have now gone seven games without a win on home soil.
“It’s disappointing because we’ve gifted them the game,” said Heckingbottom. “They’ve scored five goals from seven attempts and we’ve given them every single one of them.
“I hope we’ve made all of our errors in one game because Forest didn’t have to work hard for that.
“Forest gave us nothing – they made no mistakes and we scored two well-worked goals. We made five mistakes.
“It’s a wake-up call, massively. I think we could have won that game – when they watch it back they’ll know the same. I think we’ve been weak.”