The beleaguered Welsh side were hammered 4-0 by fellow strugglers Sunderland at the Stadium of Light on Sunday, a defeat which saw them replace Gus Poyet's men at the foot of the table.
In-form striker Connor Wickham opened the scoring and was then brought down by Juan Cala late in the first half, with the Spanish defender being sent off for the incident.
Fabio Borini tucked way the resultant penalty, despite Cardiff protests, and further goals from Wickham and Emanuele Giaccherini sealed three precious points for the hosts.
Cardiff are two points adrift of safety with matches against Newcastle United and Chelsea to come, and Solskjaer acknowledged his side are up against it, despite remaining positive.
"The effort and attitude was there for everyone to see. We go into the next two matches knowing we have to win those two," Solskjaer said.
"It's a great challenge for everyone at the club. We need two strong performances. The most we can get is 36 points and, if we can get that, I think that will be enough to keep us up.
"We've given ourselves a mountain to climb and we need to start climbing."
The former Manchester United striker refused to be drawn on his opinion on Cala's dismissal following a defeat that he admits could relegate them.
"Going in at half time 2-0 down made it very hard for us," he added.
"But I don't want to talk about the decision because it's been made and there is absolutely nothing I can say to change that.
"It could (send us down) but I don't want to comment on the decision. It's the result that matters and we lost."
