The visitors then half-heartedly appealed for a penalty when Britt Assombalonga went down after turning Gaetano Berardi on the edge of the area, but to little avail.
Leeds enjoyed their first opening on eight minutes as Jack Harrison, on his first league start for the club, attempted to play in Mateusz Klich, though Boro were able to clear.
The hosts then mustered two concrete chances in quick succession through Samu Saiz and then Berardi, the latter drifting inches wide of the post.
Neither side could break the deadlock by the half-time mark but both sets of fans were entertained as the two teams came closer and closer as the game went on.
Britt Assombalonga headed arguably the best chance of the opening period over the bar after around 25 minutes, while Luke Ayling saw a powerful header hacked off the line by Adam Clayton.
The first attack of the half fell to Boro, but Ryan Shotton's header was straight into the hands of Peacock-Farrell, while at the other end, Kemar Roofe fluffed his lines with a long-range attempt.
A number of niggly fouls punctuated the game as the minutes ticked by, with Barry Douglas and Besic both seeing yellow heading up to the hour mark and Clayton cautioned just after.
Leeds had one final chance to win the game late on when Aden Flint brought down substitute Patrick Bamford in a dangerous position, but the ensuing free kick by Douglas was easily caught by Darren Randolph as the final whistle blew on a share of the spoils.
LEEDS UNITED (4-1-4-1): Peacock-Farrell; Ayling, Cooper, Berardi (Jansson 87'), Douglas; Phillips; Harrison (Dallas 90'), Klich, Saiz, Alioski; Roofe (Bamford 90')