Mitch Clark went over for a second try on five minutes, after a deft kick from Danny Addy, the video referee confirming the score despite suspicions of offside.
Drinkwater crashed over three minutes from half-time to level the scores for Leigh, feinting and then stepping inside to avoid two Bradford players.
Two minutes into the second half and the Centurions broke the deadlock moving 16-10 in front, Adam Higson going over in the corner on the near side after a great pass over the top from Smith.
Ben Reynolds despatched a brilliant kick from right next to the touchline to complete the conversion.
In the 49th minute Reynolds scored a penalty to push Leigh 18-10 ahead after Clark was penalised for a high tackle.
At the other end, Adam O’Brien sneaked over the try line to cut the Centurions lead down to four after Leigh gave away a penalty for the markers not being square.
Kris Welham made a brilliant interception to stop Higson going over in the corner on the near side and creating a further gap between Leigh and Bradford with 68 minutes played.
The Bulls took a 20-18 lead with just over five minutes left, Addy crashing over near the goal posts before Oscar Thomas completed the conversion.
Mickey Higham clinched victory for Leigh, going over under the goal posts in the 78th minute after a neat chip from Drinkwater.