The Tigers dominated possession but struggled to shake their lowly opponents for much of the contest, eventually breaking clear in the final quarter to win 15.16 (106) to 11.6 (72) in front of 39,148 fans.
The victory was Richmond's fifth for the season, moving them back inside the top eight.
Young midfielder Brandon Ellis was the star for the winners, collecting a career-best 39 disposals, including 16 in a blistering first quarter, as he continually found space up and down the wing.
Last week's NAB AFL Rising Star nominee Nick Vlastuin continued his good form, amassing 25 touches and using the ball with precision.
Full forward Jack Riewoldt booted three goals in a good duel with Melbourne defender James Frawley, while Dustin Martin also kicked three, all after half-time.
Click here to vote for the three best Richmond players from the match
Veteran Aaron Davey, often used as the substitute this season, revelled in his opportunity to play a full game and provided a genuine spark for Melbourne.
He had 19 disposals and booted three goals in a lively display.
Defender Dean Terlich typified the Demons' strong defensive intent with an equal game-high 10 tackles to go with 21 disposals.
Five talking points: Richmond v Melbourne
Recruit Chris Dawes provided a strong target up forward in his second game of the season, taking seven marks and booting two goals.
However, as has commonly been the case, Melbourne did not have enough prolific ball winners in the midfield.
Nathan Jones' 22 disposals was the Demons' highest as Richmond had 106 more possessions in total.
Click here to vote for the three best Melbourne players from the match
Melbourne frequently found itself chasing Tiger tail early in the match, but stayed in touch through dogged defence, intense pressure and efficient ball use moving forward.
They led by two points at quarter-time and trailed by only 11 points at the start of time-on in the third quarter.
Melbourne's improved performance should ease some of the pressure on embattled coach Mark Neeld, whose future at the club faced further scrutiny on Saturday when influential club great Garry Lyon declared his position should come under "heavy review".
Lyon was on the sub-committee that hired Neeld following Dean Bailey's sacking in late 2011.
In another blow for the already injury-hit Demons, first round draft pick Jimmy Toumpas had to be subbed from the game after twisting his left ankle during the second quarter.
Players from both teams look set to be hit in the hip pocket by the Match Review Panel on Monday after an all-in dust-up on the half-time siren.
The Tigers will look to further consolidate their top eight position in next Saturday night's blockbuster 'Dreamtime at the G' clash with traditional rival Essendon.
Melbourne, meanwhile, will need to be every bit as spirited as it was on Sunday over the next three weeks, with matches against Fremantle (in Perth), Hawthorn and Collingwood.
A committed Melbourne team took to the MCG on Sunday. Picture: AFL Media
RICHMOND 2.4 7.9 12.14 15.16 (106)
MELBOURNE 3.0 7.2 10.4 11.6 (72)
Richmond: Riewoldt 3, Martin 3, King 2, Deledio, Nahas, Ellis, Vlastuin, Morris, White, Chaplin
Melbourne: Davey 3, Dawes 2, Dunn 2, Nicholson, Bail, Howe, Evans
Richmond: Ellis, Vlastuin, S. Edwards, Deledio, Grigg, Houli
Melbourne: Davey, N. Jones, Magner, Terlich, M. Jones
Melbourne: Toumpas (ankle)
Richmond: Luke McGuane subbed out for Matt White in the third quarter
Melbourne: Jimmy Toumpas (ankle) subbed out for Rohan Bail in the third quarter
Umpires: McBurney, Hosking, Foot
Official crowd: 39,148 at the MCG
The views in this article are those of the author and not necessarily those of the AFL or its clubs.