The defending champion arrived at the Liberty Stadium knowing defeat would result in its elimination from the competition, but the game ended all square after coming down to a kicking contest between Owen Farrell and Dan Biggar.
Pool leader Clermont Auvergne's shock 34-21 defeat to Northampton Saints earlier in the day had given both sides hope of reeling in the Top 14 side and the Welsh side heads to France next weekend three points adrift.
Sarries' final fixture sees them welcome Northampton, whose Clermont triumph was its first of the campaign.
Mark McCall's men led 9-6 at the interval but were dealt a blow when Billy Vunipola, who made his return from knee surgery last weekend, did not emerge for the second half due to a wrist problem, which will have England looking on nervously as the Six Nations nears.
Farrell kicked his way past Diego Dominguez into fourth on European rugby's all-time highest scorers' list during the second 40, but he was outscored after the break by Biggar as the Ospreys salvaged the draw with a late penalty.
Sarries had seen Chris Wyles sin binned after 65 minutes for a poor tackle on Sam Davies but the Ospreys were unable to make the extra man count as the decisive score evaded them.
Northampton ended its three-month wait for a Premiership win at home to Gloucester last time out and, while it had already been eliminated from Europe, its unexpected victory over Clermont blew the pool wide open.
Ulster edged in front of La Rochelle at the top of Pool 1 thanks to a 20-13 win at Ravenhill, while Wasps' hopes were dented by Harlequins' first win of the competition in a thrilling 33-28 result at The Stoop.
Wasps had James Haskell sent off for a dangerous tackle on Jamie Roberts four minutes from time, before James Chisholm crossed to complete a remarkable turnaround, with Quins having trailed 28-12 midway through the second half.
In the day's remaining match, Exeter Chiefs kept its hopes alive with a thumping 41-10 win over Montpellier that closed the gap on Pool 3 leader Leinster to four points.
The Irish province could wrap up top spot when it takes on Glasgow Warriors on Monday (AEDT).