The Boro defender was punished further for his foul, however, as Davies got up to take the penalty himself, slamming it past the keeper.
The visitors' fortunes changed just before the interval, however, as they were awarded a spot kick of their own courtesy of a handball by David Wheater.
Just as Davies did at the other end, Grant Leadbitter took on the responsibilities and made no mistake from 12 yards to level the scores at the break.
Lee came close to restoring Bolton's lead shortly before the hour mark as he fired a shot through a crowd of players, but the ball whizzed narrowly wide.
The South Korean then turned provider as he beat his marker before pulling the ball back for Liam Feeney, who put his first-time effort just past the post.
Boro were next to threaten as Bogdan was forced into a good reaction save to tip Adam Reach's effort over the crossbar.
Bogdan could do nothing about the winning goal moments later, however, as Kike raced onto a through-ball before lifting the ball over the keeper and into the net.
Jermaine Beckford came on as a late sub for the hosts and almost made an immediate impact, coming close once before finding the back of the net, only for his goal to be disallowed for a push on Wheater.
There was to be no late drama as Middlesbrough held out for the victory, leaving Bolton still searching for their first league win of the season.