Speaking after the game, Leinster head coach Joe Schmidt said: "I'm really happy with the win to be honest, because we used 26 players including a number of first and second year Academy players and I thought that they all showed a lot of character, especially in the second half.
"When you look at the calibre they were facing, such as the likes of (Michael) Lipman, (Sam) Cordingley, (Stirling) Mortlock and (Greg) Somerville, I thought that we did well to get back into the game.
"I was pleased with Berky (Mat Berquist). He's a guy who I've known since he was 15 and he's a very calm player and someone coming from the Crusaders who can bring a bit of experience to our side.
"After six weeks of pre-season we were inevitably rusty at times and we're realistic to know that we made mistakes, but it was a good win and we've something to build on for what will be a very tough Northampton side next Friday night."
The Rebels began brightly and a sweeping back-line move in the fifth minute ended in a strong tackle by Leinster winger Darren Hudson on the advancing Richard Kingi.
Eventually the pressure from the Rebels told and an infringement at a ruck three minutes later led to a penalty in front of the posts. Former Australia captain Mortlock had little difficulty in slotting home at the Wesley end of the ground to make it 3-0 to the visitors.
Leinster rallied through the industrious Brendan Macken whose break into the Melbourne half resulted in a 12th minute penalty.
Debutant out-half Berquist opted to kick to the corner. From the resulting lineout, Leinster pressed but a spilled catch by Dominic Ryan metres from the try-line denied the hosts their opening points.
In the 24th minute, Melbourne winger Cooper Vuna instigated a move which full-back Kingi finished off in the corner. Mortlock's touchline conversion attempt fell short of the posts but it pushed their lead to 8-0.
From the restart again Leinster signalled a strong intent with stand-in captain Rhys Ruddock leading the charge into the Melbourne 22.
With Berquist an assured presence at out-half, Leinster looked to regain the iniaitive but it was his half-back partner John Cooney who launched Leinster's first notable break.
He cleverly kicked ahead on the half hour mark to the onrushing Fionn Carr but a cruel bounce of the ball robbed Carr of a certain five-pointer.
With their province improving with each passing minute, a try seemed inevitable. Cooney, Leo Auva'a and Mark Flanagan probed separately and their carries led to a penalty.
Melbourne lost their number 8 Tim Davidson to the sin-bin and with two minutes to go before the break, Ruddock capitalised on the one-man advantage to crash over in the corner for Leinster's opening try.
Berquist expertly converted to give the visitors a sender one-point lead. But right on the stroke of half-time, Melbourne lock Hugh Pyle broke free to touch down for an unconverted try, leaving the scoreline at 13-7 at the interval.
Leinster summer signing Steven Sykes entered the fray after the resumption and the province gained the upper hand early on.
Jamie Hagan and Heinke van der Merwe punched holes in the Rebels' defence as the home side forced the Rebels on to the back foot.
Eventually the pressure resulted in a crucial 52nd minute score. A superb scything Carr burst set up the match-winning try for fellow winger Hudson, who displayed impressive speed and strength to touch down under pressure in the corner.
Berquist added the difficult conversion to give Leinster a slender 14-13 advantage. With both teams opting to make a series of substitutions the game see-sawed, but the men in blue deservedly edged a competitive contest, which saw a number of inexperienced players impressing, in front of a crowd of 3,520.
LEINSTER: Ian Madigan; Darren Hudson, Brendan Macken, Eamonn Sheridan, Fionn Carr; Mat Berquist, John Cooney; Heinke van der Merwe, Richardt Strauss, Jamie Hagan, Mark Flanagan, Devin Toner, Rhys Ruddock (capt), Dominic Ryan, Leo Auva'a.
Replacements used: Steven Sykes for Flanagan (half-time), David Gannon for Toner, Andrew Conway for Madigan (both 54 mins), Cillian Willis for Cooney, Ben Marshall for Auva'a (both 60), Aaron Dundon for Strauss (61), Ian Madigan for Berquist, Jack McGrath for Hagan, Ciaran Ruddock for Ryan (all 69), Andrew Boyle for Hudson, Jack O'Connell for van der Merwe (both 74), Colm O'Shea for Sheridan, Dominic Ryan for Ruddock (both 79). Not used: Jason Harris-Wright.
MELBOURNE REBELS: Richard Kingi; Mitch Inman, Stirling Mortlock, Lloyd Johansson, Cooper Vuna; James Hilgendorf, Sam Cordingley; Nic Henderson, Heath Tessman, Greg Somerville (capt), Hugh Pyle, Luke Jones, Jarrod Saffy, Michael Lipman, Tim Davidson.
Replacements used: Eddie Aholelei for Henderson (15 mins), Ged Robinson for Davidson (37-47, temp), Peter Betham for Inman, Lachlan Mitchell for Johansson (half-time), Rodney Blake for Aholelei (49), Danny Cipriani for Hilgendorf (54), Cadeyrn Neville for Saffy (55), Laurie Weeks for Somerville (58), Mitch Inman for Mortlock (59), Pama Fou for Cordingley (61), Lloyd Johansson for Vuna (71), Luke Holmes for Lipman, Nigel Ah Wong for Davidson (both 72), Dave Hardisty for Kingi, Liam Windon for Betham (both 76). Not used: Ben Matwijow.
Referee: David Wilkinson (IRFU),