Allardyce’s side recovered from an awful start during which Benik Afobe gave Bournemouth the lead with a 13th-minute header – their eighth goal inside the opening 15 minutes this season.
The early setback dented the Black Cats’ fragile confidence further, but they gained a foothold when full-back Patrick van Aanholt fired home an equaliser in first-half injury time.
Billy Jones would have given the home side the lead after the break but for a fine save by Artur Boruc, and substitute Lewis Grabban passed up a glorious opportunity to win it for the visitors late on.
Asked if Sunderland might have collected all three points if they had gambled, Allardyce was pragmatic.
“Why? Why gamble when we have done it before and lost every single game? You don’t gamble on situations like that, particularly when you have gambled before and ended up losing three and four and not gaining any points whatsoever.
“This is the first point we have gained from going a goal down since I have been here.
“Winning a game from going a goal behind is something this club doesn’t do very, very well if you look at the history of it, so to actually get a point is a good achievement from going a goal behind.”
Opposite number Eddie Howe was disappointed his men had not made the most of their early dominance, and although they sit four points clear of the drop zone, he is not content with their current situation.
Howe said: “That’s a difficult one. From my perspective, I always want more and I set such high standards for the players that it’s difficult to settle for where we are now. That’s a tough question to answer.
“I think I’d have liked to have seen us higher.”
Bournemouth’s latest point came at an added cost with midfielder Junior Stanislas waiting to discover the extent of suspected medial ligament damage.
Mannone, Jones, O’Shea, Brown, Van Aanholt, Cattermole, M’Vila, Johnson, Lens (Rodwell 88), Borini (Watmore 66), Defoe.
N’Doye, Pickford, Coates, Yedlin, Fletcher.
Boruc, Smith, Francis, Cook, Daniels, Surman, Stanislas (Murray 90), Gosling, Arter, Pugh (Ritchie 57), Afobe (Grabban 65).
Iturbe, Federici, Distin, O’Kane. Ref: Roger East (Wiltshire).