Alexander Anyukov squeezed a shot in off Oliver Kahn's legs to give Russia the lead in the 26th minute before Schweinsteiger got off the mark, finishing at the far post to level after half an hour.
Schweinsteiger, the 20-year-old Bayern Munich winger, swept in right-footed with 20 minutes to go but Germany were denied victory when Alexander Kerzhakov struck in the first minute of added time after a swift break from Andrei Arshavin.
"The combination play was very nice to look at," said Germany coach Juergen Klinsmann, who was not too concerned by the late equaliser.
"We're in a process of fine-tuning at the moment."
Germany, who beat Northern Ireland 4-1 in Belfast on Saturday, took time to get going against Russia, with only the running of Andreas Hinkel causing the visitors any problems.
Russia, boosted by a 2-0 Group Three win over Latvia on Saturday that keeps them in contention for a return to Germany in next year's World Cup, almost had an opening goal in the sixth minute when Dmitry Loskov let fly from just outside the box.
Sergei Semak, on as an early substitute after Marat Izmailov had been forced off by a Michael Ballack tackle, came even closer when his long-range effort brought a fine save from Kahn at full stretch.
Russia were in control of midfield and they moved ahead when Anyukov exchanged passes with Dmitry Sychev on the edge of the area on the right and slipped his shot past the Bayern Munich keeper.
Germany's equaliser came from another of Hinkel's forward bursts.
The VfB Stuttgart defender cut in from the right and crossed low into the penalty area, where Lukas Podolski got an important touch to set up Schweinsteiger to send in a deflected shot at the far post.
Podolski twice went close with first-time strikes from good positions before Schweinsteiger stepped over the ball, weaved infield and unleashed his low shot for the second.
Germany continued to press and Ballack had a shot saved at point-blank from Sergei Ovchinnikov in the 78th minute.
A home win looked certain before Arshavin broke clear down the right, rounded Kahn and crossed low for Kerzhakov to slide in and score the equaliser.
Germany play Australia, Argentina and Tunisia in Group A of the Confederations Cup, a dress rehearsal for next year's World Cup.
World champions Brazil are in Group B along with Mexico, Japan and European champions Greece.