Trinity's ability to score long-distance tries proved crucial on an afternoon of dazzling attack and leg-weary defence.
"We are still in a relegation fight and our livelihoods are at stake, but the pride and commitment all 17 players showed was magnificent," said Smith, who has now had two of the three games he was given to prove he should get the job.
Wakefield showed more hunger than the Bulls, whose coach, Brian Noble, admitted: "Too many of our players took short cuts.''
Bradford took the sort of early lead that suggested that it might be a routine afternoon, Joe Vagana getting a clever pass away and Paul Deacon sending Michael Withers over.
Throughout the game, though, Wakefield were at their most dangerous when penned in their own half. Sid Domic led the breakout and Duncan MacGillivray went over for the equalising try. They showed the same characteristic even more clearly when Colum Halpenny fielded Deacon's kick near his own line and raced the length of the field to score.
Trinity were as fallible in defence as they were exciting in attack, with Deacon finding a huge gap to reduce their lead. It was then Leon Pryce's turn to have his kick returned with interest, Semi Tadulala starting the counter-attack but leaving Ben Jeffries with plenty to do for the try that restored Wakefield's eight-point lead.
It all went wrong for them in the minutes before half-time, Karl Pratt twice scoring tries far too easily, and Iestyn Harris involved in creating both.
The game could not quite sustain its breathless rhythm after the break, but Trinity regained the lead early in the second half when Michael Korkidas powered over. Pratt completed his hat-trick from Withers' kick, but it was Trinity who finished the fresher.
Their man of the match, Jeffries, found the gap to put them back in front and then the late introduction of Jamie Rooney proved an inspired move. He scored with 10 minutes to play and, even though Brad Myers replied for Bradford, Rooney found touch with his kick-off and grabbed a second try in injury time.
Wakefield: Halpenny, M Field, Demetriou, Domic, Tadulala, Jeffries, Obst, Griffin, March, Korkidas, Solomona, MacGillivray, J Field. Substitutes used: Rooney, Snitch, Elima, Wrench.
Bradford: Withers, Reardon, B Harris, Johnson, Pratt, I Harris, Deacon, Vagana, Henderson, Fielden, Peacock, Myers, Radford. Substitutes used: Pryce, Langley, Parker, Lynch.
Referee: K Kirkpatrick (Warrington).
* Huddersfield's stand-in captain Chris Thorman scored 20 points in his side's 32-22 win over struggling Leigh to keep them in touch with the play-offs. Thorman, leading the side in the absence of the injured Ben Roarty, scored a try and kicked eight goals to all but condemn Leigh to the drop.Reuse content