Forest goalkeeper Paul Smith picked up the man of the match award for a series of magnificent saves to deny Swansea the chance to climb back into the play-off places.
Swansea started brightly and looked determined to prove their worth as promotion contenders but found Smith on inspired form.
A low drive from striker Lee Trundle after only four minutes was kept out by the feet of Smith.
Fellow frontman Adebayo Akinfenwa should have put Swansea into the lead on 23 minutes but headed a Leon Britton cross directly into the arms of the keeper.
Forest's powerful forward Junior Agogo broke free after 36 minutes but Swansea goalkeeper Willy Gueret was quickly down at his feet to stifle the threat.
Both sides created a number of half chances in a lively first period and Ian Craney almost gave Swansea the lead right on the half-time whistle but his 25-yard drive was expertly saved by Smith.
Smith was called into action almost immediately in the second half when Trundle rifled a volley from 25 yards which the keeper did well to push over the bar.
Forest striker Grant Holt slid in at the back post to meet a Nicky Southall cross on 52 minutes but his left-foot shot was poor.
Nathan Tyson's pace began to cause the Swansea back four some problems in the second half. He found himself one-on-one with Gueret after 76 minutes but his chip was well held by the Swansea keeper.
Swansea pressed for a winner and came close on 81 minutes when Britton's cross from the right found Akinfenwa but his header was magnificently saved.