Midfielder Mellish, who had only previously scored twice all season, did the damage as Paul Simpson’s basement boys bagged just a sixth win of a torrid campaign.
Mellish rose unchallenged to meet substitute Jordan Gibson’s cross after Posh switched off from a Dylan McGeouch short corner as Carlisle took a 27th minute lead.
The same man doubled their advantage three minutes into the second half when he struck an impressive volley past Posh goalkeeper Jed Steer, following a weak defensive header from England Under-20 defender Ronnie Edwards.
Mellish was not done there, completing a memorable treble in the 58th minute with another spectacular left-footed volley after the hosts failed to deal with a McGeouch cross.
Posh quickly hit back as captain Harrison Burrows slammed a 61st-minute shot through a crowd of players from the edge of the box but that was as good as it got as Darren Ferguson’s men flopped.
Substitute Jonson Clarke-Harris fluffed a glorious chance to cut the arrears further, before also being denied by the bar late on as the courageous Cumbrians stood firm.
The Blues qualified for the play-offs after finishing the regular campaign in fifth position and came from behind in the tie against Bradford to book their place at Wembley, with a 3-2 aggregate win.
Simpson took over at Brunton Park in February 2022 with the club battling against relegation to the National League and has gone on to lead them to within one game of promotion.
The 56-year-old is aiming for his third promotion with the Cumbrian club, having been player-manager from 2003 to 2006, but admitted a lot of work needed to be done when he walked through the door.
Simpson told the PA news agency: “No, I did not think we would have been in this position.
“There were lots of things that needed to be changed. I didn’t think it was an absolute disaster but certain issues needed to be resolved.
“It was a big challenge to try bring all the different factions of the football club back together – there was a big difference between the office staff and players, there was not a good feeling with that, dislike towards the directors, no local businesses involved. I wanted to bring everyone back together.
“Everyone has pulled together over the last five home games where we are getting big crowds and that has been really satisfying.”
The Blues come up against Stockport, who are aiming to make it back-to-back promotions after they were crowned champions of the National League last season, and will aim to do what Simpson did with Carlisle in his previous spell in charge, taking them from non-league to the third tier in consecutive years.
The United boss knows his side were under-estimated heading into the post-season, but has acknowledged the size of the task in hand if they are to stop Stockport – a side that finished the season unbeaten in 13 before coming from behind to beat Salford in their respective play-off semi-final.
Simpson continued: “Some of our local media were saying that we are the team that everyone wanted to play.
“I find it disappointing and disrespectful to the players that have been outstanding all season.
“We haven’t achieved anything apart from getting to Wembley and now we have to try and get a promotion. When you get to a play-off final there is nothing for coming second.
“I’m sure the stadium and surroundings will get to the players – I know from experience what Wembley does to players and staff so it’s our job to help each other.”
Callum Guy helped put Carlisle in front with a terrific strike in extra-time last weekend against one of his former teams, scoring just his fourth goal in Blues colours.
The 26-year-old is aiming to win his first promotion as a player.
He said: “The weekend can’t come quick enough.
“I’ve not scored a league goal before this year but I’ve managed to score four this year. I think I’ve just taken a few more risks this year, obviously it helps when my team is doing really well.
“We’ve showed over the two legs against Bradford, with the crowds that have been there, we can handle the big occasion and we can score goals in them.”
