The 33-year-old took his league tally to 15 goals after firing Werder into a 29th minute lead and restoring their one-goal cushion two minutes into the second half after Freiburg's Cedric Makiadi had grabbed a 32nd minute equaliser.
Werder could have extended their lead in the 53rd but Markus Rosenberg's shot rattled the post and Jonathan Schmid levelled again for the hosts in the 71st, beating goalkeeper Tim Wiese from a tight angle, with the ball rolling off the post into goal.
Werder, who have drawn their last three games since the Bundesliga restart following the winter break, move up to 32 points, one ahead of Bayer Leverkusen.
"We wasted two points today and this is the reason I am a bit angry," Pizarro, the Bundesliga's all-time foreign goal scorer, told reporters.
"We needed these points so that the teams ahead of us don't pull away too much. But we lacked concentration at some instances, especially set pieces."
In the only other game on Sunday Cologne, without their injured top scorer Lukas Podolski, snatched a 1-0 win at relegation-threatened Kaiserslautern and three crucial points in their bid to stay up.
Albania international Odise Roshi's 72nd minute header lifted them to ninth place on 24 points.
Kaiserslautern, reduced to 10 men late in the first half after the dismissal of new signing Ariel Borysiuk for a second booking, are in the relegation play-off spot on 18, a point off the bottom.
Champions Borussia Dortmund have opened up a two-goal lead at the top of the table after their 2-0 win at Nuremberg on Friday with Bayern Munich stumbling to a 1-1 draw at Hamburg SV a day later to drop to second place on 41 points.
Third-placed Schalke 04 also settled for a 1-1 draw against Mainz 05, also on Saturday, to join Bayern on 41.
Get the best features, fun and footballing frolics straight to your inbox every week.
Thank you for signing up to Four Four Two. You will receive a verification email shortly.
There was a problem. Please refresh the page and try again.