The goalless draw at the City Ground means it is all still to play for in the second leg in south Wales on Monday night.
A sensational start to the game saw Taylor dismissed after only 52 seconds.
The Swans defender caught Lewis McGugan on the thigh with his studs as both players went into the challenge with raised feet just inside the Swansea half on the left touchline.
McGugan got to the ball first and, with the home crowd baying for blood, referee Mike Dean pulled out his red card.
Swansea boss Brendan Rodgers was forced into an unexpectedly early reshuffle. Stephen Dobbie was the player sacrificed as Garry Monk was introduced from the bench in the fourth minute.
However, while Forest struggled to break down 10 men, the Welsh side were unfazed by their numerical disadvantage and created the better chances in the first half.
Fabio Borini was the main threat. The on-loan Chelsea forward forced Forest goalkeeper Lee Camp into saves on three occasions.
In the 20th minute he beat Luke Chambers for pace down the left side, cut into the penalty area and Camp had to rush from his six-yard box to smother the danger.
Soon afterwards, he went even closer with a superb 25-yard free-kick.
His run was almost Cristiano Ronaldo-esque and his shot was not too unlike that of the world's most expensive player either as he struck the ball over the wall and forced Camp to help it over the bar.
Borini worked Camp again on the half-hour, shooting through Brendan Moloney's legs after being slipped in down the left side of the area by Nathan Dyer. Camp blocked well with his legs after seeing the ball late.
It took Forest 36 minutes to muster an effort of their own.
Kris Boyd got free down the right-hand side. His low cross was met by the penalty spot by Robert Earnshaw, who controlled his half-volley with the side of his foot but Dorus De Vries dived to his right and palmed the ball away.
Chris Cohen's corner was headed over by Chambers at the back post on the stroke of half-time.
Forest continued after the break where they had left off in the first half.
A dipping 25-yard shot from McGugan was palmed back into the danger area by De Vries before the ball was cleared by a defender.
McGugan then produced a defence-splitting pass to pick out Cohen, but his pull-back to the edge of the box was blasted over by Earnshaw in the 55th minute.
McGugan struck another long-range effort on the hour, but this time De Vries watched the ball skid off the greasy surface and made a routine save.
At the other end, Camp did likewise after Scott Sinclair failed to get hold of an effort on the run.
Forest thought they had made the breakthrough after 68 minutes when Earnshaw headed home McGugan's corner, but the Wales international was flagged offside.
Swansea immediately went on the counter-attack. Dyer darted down the right side and struck an early shot from the edge of the area that brought a superb one-handed save from Camp as he got down to his left.
Forest substitute Nathan Tyson saw a goalbound effort blocked after dancing past a couple of challenges 18 yards out. The speedy forward was then inches away from connecting with a Cohen cross as he and Paul Anderson stretched every sinew in an attempt to get a telling touch on the ball.
Anderson then saw his 81st-minute effort from 25 yards pushed wide by De Vries.
Forest continued to search for a crucial goal to take to south Wales but it was not forthcoming, with Chambers heading harmlessly over from another Cohen cross in injury time.