The Red Devils had seen off Perth Glory, Leeds United, Inter Milan and Tottenham Hotspur ahead of this friendly at the Ullevaal Stadion, but they struggled to find a way through in Tuesday's match.
Kristiansund, who brought on Solskjaer's son Noah late on, were unable to hold on for a famous result as second-half substitute Mata won a penalty that he successfully converted late on.
Kristiansund boast the best defensive record in the Norwegian top flight and that showed in the first half as United failed to find a way through.
United grew more and more into the game as first half wore on, though, and if not for some fine goalkeeping from Sean McDermott they would have been ahead at the interval.
McDermott was equal to a couple of Marcus Rashford attempts, including a 25-yard free kick, and he also denied Martial from a promising position.
With the pressure building at the end of the first period, McTominay fired a shot inches over the bar and then thumped his next attempt against the frame of the goal.
Solskjaer opted against making any changes at the interval but his side picked up from where they left off as Rashford's shot was saved by substitute keeper Serigne Mor Mbaye 30 seconds into the second half.
Mbaye also thwarted another Rashford shot on the hour, this time from further back, and Daniel James was likewise blocked, then United were fortunate not to concede when Sondre Sorli headed wide from a glorious position.
The Spanish midfielder latched on to Paul Pogba's through-ball and was brought down by Andreas Vaikla after rounding the substitute goalkeeper, gifting him the chance to keep United's momentum going by tucking away the resulting penalty.
KRISTIANSUND (4-3-3): McDermott (Mbaye 46; Vaikla 87'); Coly (Williamsen 87'), Ulvestad (Sormo 46'), Psyche (H. Gjesdal, 67), Aasbak (Sorli 46'); Cirak (Hoven 67'), Diop, Kalludra (Sivertsen 46'); Bye (Olsen 65'), Kastrati (Amang 46; Solskjaer 87'), T. Gjertsen (Isaksen 70')
MAN UNITED (4-2-3-1): De Gea (Romero 60'); Wan-Bissaka (Dalot 60'), Jones (Tuanzebe 60'), Lindelof (Greenwood 60'), Shaw (Young 60'); Matic (Pogba 60'), McTominay (Rojo 60'); James (Fred 60'), Lingard (Chong 60'), Martial (Mata 60'); Rashford (Pereira 60')