The point proved enough to nudge the Bluebirds above relegation rivals Fulham - who lost earlier at Tottenham - but they remain in the drop zone with just three games of the season remaining.
The home side had made a superb start with both Mats Daehli and Jordan Mutch missing fine chances in the early stages before 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.
The home side thought they had gone ahead when Juan Cala prodded home on the hour but his effort was ruled out for offside and Stoke could have won it at the death with Oussama Assaidi bringing a brilliant save out of David Marshall and Jon Walters hitting the bar with two minutes to go.
Solskjaer could seek some solace from Fulham's 3-1 defeat to Tottenham at White Hart Lane.
Christian Eriksen was the Spurs star as he set up two of the goals which brought an end to a run of two straight victories for Felix Magath's battling Cottagers.
But it was far from straightforward for the home side had who had goalkeeper Hugo Lloris and some wayward finishing to thank for restricting the visitors to one Steve Sidwell goal.
Sidwell had fired a 37th-minute equaliser just two minutes after Paulinho had turned home an Eriksen free-kick to give his side the lead.
Harry Kane glanced home an Aaron Lennon cross three minutes into the second half and Younes Kaboul sealed the points for Spurs by knocking in another Eriksen free-kick just past the hour mark. Sidwell then saw a penalty saved by Lloris.
Wilfried Bony all but sealed Swansea's survival and piled more misery on Newcastle boss 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 grabbed the winner from the spot in injury time after Marvin Emnes was brought down by Cheick Tiote in the box, lifting the delighted 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 uncharacteristically missing two early chances for Saints, while Villa's best chance came from Ashley Westwood who blazed over in the second half.
Crystal Palace claimed their fifth straight victory in an emotional 1-0 win over West Ham at Upton Park.
The match was preceded by a minute's silence for Hammers 20-year-old Dylan Tombines who lost his battle against testicular cancer on Friday.
Mile Jedinak won the game with a 59th-minute penalty after Cameron Jerome was brought down by Pablo Armero in the box.