The point nudged the Bluebirds above relegation rivals Fulham - who lost at Tottenham - but they remain in the drop zone with just three games left.
Marko Arnautovic put the visitors in front from the spot on the stroke of half-time. Peter Whittingham levelled matters in the 51st minute, also from the spot, after Fraizer Campbell was fouled by Steven Nzonzi in the box.
Solskjaer could take some solace from Fulham's 3-1 defeat at White Hart Lane. Christian Eriksen set up two of the goals to help beat Felix Magath's battling Cottagers.
But it was far from straightforward for the home side. Goalkeeper Hugo Lloris and some wayward finishing was all that restricted the visitors to one Steve Sidwell goal.
Wilfried Bony all but sealed Swansea's survival and piled more misery on Newcastle manager Alan Pardew by scoring both goals in a 2-1 win at St James's Park.
Shola Amoebi fired home in the 23rd minute to give the Magpies their first goal in four games, but the discontent grew when Bony headed in a Ben Davies cross to equalise on the stroke of half-time.
Bony got the winner from the spot in injury time after Marvin Emnes was brought down by Cheick Tiote, lifting the Swans to 36 points and handing Pardew his fifth successive defeat.
Aston Villa are still not assured of safety despite ending their run of four straight losses with a goalless draw against Southampton at Villa Park.
It could have been worse for Paul Lambert's men with Rickie Lambert missing two early chances for Saints. Ashley Westwood had Villa's best chance in the second half but fired over.
Crystal Palace claimed their fifth straight victory beating West Ham 1-0 at Upton Park with a penalty from Mile Jedinak in the 59th minute.