Danjuma netted in the second half to the delight of around 2,000 Bournemouth fans allowed into Dean Court after coronavirus restrictions were reduced in England this week.
Bournemouth are bidding for an immediate return to the Premier League following last season's relegation.
They went into the play-offs in poor form, losing their final three games to finish sixth in the second tier, 10 points behind Brentford.
Cherries boss Jonathan Woodgate claimed his side were the underdogs going into the play-offs.
But now they need only avoid defeat in Saturday's second leg to reach the final against Barnsley or Swansea, who play their first leg later on Monday.
Brentford are battling to avoid another play-off letdown after decades of failure in the end of season lottery.
Thomas Frank's team had won four in a row in a 12-match unbeaten run at the end of the regular season, securing a second successive third place finish.
However, Brentford have taken part in a record nine play-off campaigns without winning promotion.
The west London club have lost four play-off finals, including last season's agonising 2-0 defeat against local rivals Fulham at Wembley.
Bournemouth had the better of a cagey first half and Dominic Solanke headed against the outside of the post from a Lloyd Kelly cross.
Philip Billing skewed wide when the ball fell kindly for the unmarked Bournemouth midfielder in the penalty area.
Brentford keeper David Raya kept out Steve Cook's header just before half-time, with Ivan Toney on hand to block David Brooks' follow-up effort.
Bournemouth's pressure paid off 10 minutes into the second half as Dutch winger Danjuma started and finished a clinical counter-attack.
Danjuma helped dispossess Bees defender Pontus Jansson and, after the ball was worked on by Solanke, he raced on to Brooks' through pass and slotted past Raya.