The Bluebirds failed to capitalise on a dominant first-half display at Cardiff City Stadium, hitting the woodwork through Fraizer Campbell and a deflected Craig Noone strike.
And in the end they were again indebted to goalkeeper David Marshall, who made two stunning late saves to deny Fabien Delph and Andreas Weimann to ensure the game ended goalless.
Cardiff's season has been a worrying combination of a failure to score goals and an inability to keep them out at the other end.
The Bluebirds have scored the second lowest number of goals in the top flight, and conceded the second highest tally.
They have now failed to score in 14 of their 26 league games, and the shut-out against Villa was just their sixth this term, and their first under Solskjaer.
But the Norwegian hopes an improvement at the back will ultimately prove the foundation for escaping the drop and an immediate return to the Championship.
"Ben Turner and Steven Caulker did as well as you can ask from two centre-backs in this game, thats's where you take lots of encouragement from," he said.
"We had a solid foundation and that is what we have spoken about. Clean sheets give you a foundation to play off and that is more vital in this situation than anything else, and it gives you a platform to win games and you saw as a result we had more than enough chances to beat Villa."
Solskjaer acknowledged the stalemate was a chance missed for Cardiff, who remain 19th in the table, but is confident his side will pick up points in their upcoming home games.
"It is at home that we are going to take the majority of the points that we need to stay up," he said.
"We have some very exciting games coming up here at home, and there are not a lot of teams in the Premier League who are better than Villa away from home, so when you look at our home games this was maybe the most difficult one."
The Norwegian also asked for the fans to be patient with loan signing Wilfried Zaha, who struggled to make an impact against Villa.
"You have to remember he has not played much this season, pretty much since pre-season," said Solskjaer of the Manchester United winger.
"I am sure he will be better for the three games he has had."
Villa boss Paul Lambert will use his side's lack of a game this weekend, having been knocked out of the FA Cup, to take his players away for warm-weather training in Portugal.
The Scot hopes it will leave his players refreshed for the final run-in.
He said: "It's a change of scenery, we will still train every day, we will be there to work. It will be like a normal week but with no game on Saturday. I have done it before with teams and it is good for the players."
Villa are five points ahead of the drop zone and, while Lambert knows they are not out of the woods yet, he is content with their position compared to the situation they found themselves in a year ago.
"There are too many games left to play, but the thing with us is we are a threat to teams and we can score, as you saw when we had those chances to win this game," he said.
"We will keep going, we were in a hell of a position this time last year and we got out of it so it is too early to judge it yet."