The Bees never really recovered after conceding two goals in the first seven minutes.
Curran, 17, was making only his second start for the Merseysiders, but struck in the fifth minute with an excellent right-foot shot on the turn, after an effort by Calvin Zola had been blocked.
Curran netted again in the seventh minute, this time with his left foot, although the ball took a deflection to deceive Brentford keeper Nathan Abbey.
It was all Tranmere for the first half hour.
Chris Greenacre got in the action after only 16 minutes when replacing the injured Zola and was unlucky on a couple of occasions not to register his 20th of the season.
Brentford looked to have got back in the game after 35 minutes when Calum Willock headed in after a corner, but just 60 seconds on it was 3-1 for Tranmere.
Curran latched on to a pass from Greenacre after a long clearance by keeper Ben Hinchcliffe and netted with an excellent volley over Abbey.
The second half was something of a disappointment but Brentford never threw in the towel. Substitute Alex Rhodes made an impact and was only inches off target with two good shots from the edge of the box.
Charlie Ide was Brentford's best on the day but was forced off with a knee injury after 75 minutes.
Curran took star billing for Tranmere but there was also some good work from John Thompson, Danny Harrison and Steve Davies.
It was Tranmere's 13th home win, but still left them bitterly disappointed after failing to make the play-offs when they had looked in with a great chance early on in the New Year.