Connor Wickham gave Sunderland the lead within the opening minute after capitalising on an error by Stoke keeper Asmir Begovic, but Adam netted a superb leveller before half-time.
The visitors held on for the point but results elsewhere saw them drop into the precarious position of 18th with just five games left to go this season.
Sunderland had made the briskest of starts, Wickham putting them ahead in the opening minute.
Will Buckley got down the right for the Black Cats before whipping one straight into the arms of Begovic.
It was the kind of effort the Stoke keeper would usually easily smother nine times out of ten, but he somehow managed to let it squirm out of his grip and into the path of Wickham – who took a touch before toe-poking in from close-range.
Stoke responded well, however, and took the game immediately back to Sunderland. First, Jonathan Walters had a low effort smartly saved from the edge of the area, before Stephen Ireland’s long-range volley was nearly deflected in by a defender.
But it was Sunderland who had the next genuine opportunity to score, with Jermain Defoe the guilty party as he fired into the side netting from a position in the area where he really should have done better.
And the former Tottenham striker was made to rue his miss just moments later when Adam levelled the game.
The Scot may be more well-known for his left foot, but he proved he was just as capable with his right by driving one home into the top right-hand corner after the ball had dropped back to him on the edge of the area.
Stoke defender Phillipp Wollscheid may consider himself a little lucky to have escaped red just before the interval, after he caught John O’Shea full in the face with studs as the Irish defender went in for a header.
There were chances at both ends as the second-half progressed. In quick succession Billy Jones charged into the area before seeing his shot saved impressively by Begovic, before Walters headed over from close-range at the other end when he probably should have done better.
Mame Biram Diouf then had a real chance to put Stoke in front after the ball had found him from a Adam free-kick, but his effort was saved superbly at point-blank range by Costel Pantilimon in the Sunderland goal.
Adam, ever a threaten from long-range, then had a chance to win it late on, but his shot that seemed destined for the top corner was just tipped wide by the Black Cats keeper.
A point at the Britannia would never usually be a bad result for Sunderland, but Dick Advocaat’s side now find themselves in real trouble after wins for relegation rivals Hull and Leicester City.
Paul Merson’s Soccer Saturday verdict
Sunderland have been entertaining. A prime example was, when they were walking off, they were getting clapped. They’ll give it a right go. It will be open, it will be entertaining and nervy but that’s the way they’ve chosen to play. Charlie Adam was outstanding, Arnautovic was a handful. It was a very good game all round.
Stoke: Begovic (4), Cameron (5), Shawcross (6), Wollscheid (6), Pieters (6), N’Zonzi (6), Adam (7), Ireland (6), Walters (6), Arnautovic (6), Diouf (5)
Subs: Bardsley (5), Odemwingie (5) Crouch (5)
Sunderland: Pantilimon (7), O’Shea (7), Coates (6), Jones (6), van Aanholt (6), Cattermole (6), Rodwell (5), Gomez (5), Wickham (6), Buckley (5), Defoe (4)
Subs: Graham (5)
Man of the match: Charlie Adam