The Tigers' 14.11 (95) to 11.11 (77) win is their sixth straight and has them sitting just a game outside the top eight.
Meanwhile, the Bombers, who are now just one game clear of the Tigers, could fall out of the eight by the end of round 20 depending on the results in North Melbourne, Collingwood, Gold Coast and Adelaide's games.
The Tigers started the game slowly, turning the ball over time and time again in the first term to enable the Bombers to go into quarter-time 12 points up.
But Richmond started to take advantage of their dominance at stoppages in the second term, lifting their pressure around the ground to deny Essendon the short options they had used so effectively in the first term.
The Tigers piled on six goals to Essendon's three in the second term to take a six-point lead into half-time, setting up a classic contest in the second half.
The momentum swung again at the start of the third term, when Essendon rediscovered its run to pile on the first three goals of the second half to grab an eight-point lead.
But the Tigers kicked the next four goals, taking a handy 18-point lead at the three-minute mark of the final term when Nathan Gordon converted with a kick that sliced through from 30m out.
The Bombers were far from done, kicking three of the next four goals and closing to within five points at the 20-minute mark when Dyson Heppell kicked a behind.
However when Shane Edwards kicked a goal three minutes later to put the Tigers 12 points up, Richmond was able to hold on despite some tense moments.
Richmond coach Damien Hardwick said he was really proud of his team's effort.
"They were hard and tough and Essendon challenged us at the end there and we managed to bounce back, which was really impressive from our guys," Hardwick said.
"Defensively we were a lot better after quarter-time and we managed to keep the score down, but they control and move the ball so well and we probably defended a little bit much from the back half.
"I think we had nine forward-half turnovers, which is well below our season average, so we'll look at some things and see if we can do some things better moving forward."
A frustrated Essendon coach Mark Thompson said he didn't have any answers for his team's costly loss.
"It's a pretty disappointing night for the football club," Thompson said.
"It's hard to be positive when you lose a game to Richmond, who wanted to win more than us to play finals.
"In reality we played good football in bits and pieces. The first quarter was OK, towards the end of the first quarter we went away from how we want to play and then you could just see it coming.
"And we were too easy to play against, clearly."
Former Giant Anthony Miles continued his recent run of good form for the Tigers, finishing with 28 possessions and a game-high seven clearances.
Brandon Ellis (28 possessions) was a key playmaker for the Tigers all night, Edwards (19 possessions and two goals) was lively up forward and in the midfield, while Troy Chaplin did an outstanding job in defence on Joe Daniher.
Alex Rance had an entertaining duel with Bomber Jake Carlisle. Carlisle's height troubled the Tiger full-back at times but he battled hard to keep the Bomber to two goals.
Dyson Heppell and David Zaharakis (both 32 possessions) were outstanding for the Bombers around the ground, while full-back Cale Hooker continued his outstanding 2014 form keeping Jack Riewoldt to two goals.