The 20-year-old Norwegian shot stopper arrived in West Yorkshire over the summer on a four-year deal to become Illan Meslier's backup under head coach Marcelo Bielsa in LS11.
Klaesson has been a regular for the Whites Under-23s outfit this term since making the switch from Eliteserien side Vålerenga.
Sporting director Victor Orta had been tracking the young goalkeeper for a number of years before eventually securing his services in the wake of Kiko Casilla's departure back to Spain on loan.
Klaesson has made nine appearances in total for Jackson's development side so far, though has only kept one clean sheet amid an early season leaky defence that has conceded 18 Premier League 2 goals.
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The Whites were beaten 4-2 at Everton on Monday night with United's second choice in between the posts again gaining more valuable on-pitch minutes.
Klaesson made a fine double save in the first half in Southport - the second of which was a brilliant palmed stop onto the crossbar from close-range - having earlier provided a pinpoint pass forward for Amari Miller to open the scoring against the Toffees.
His adjustment, though, will take time and understanding to get fully up to speed as the club's number two.
"Like any new player there's a settling in period for them," Jackson told the YEP of Klaesson's first few months.
"There's an adaptation to the way we play, particularly for a goalkeeper coming into what could be new training methods. As with outfield players it's a new way of playing and it's the same with goalkeepers.
"He's a player who the staff are working really hard with to help him adapt as quickly as possible. There was a great double save from him [against Everton] and the ball forward for Amari's goal was a pass forward [not an aimless clearance].
"He saw it and we're pleased for him in that respect but he won't be happy conceding four goals and we're not as a team. In regards to him settling in, he is slowly getting used to it and he's coming along nicely."