After a turgid first half which saw Gareth Steenson and Mike Delany trade penalties, the second half burst into life as Chiefs' Thomas Waldrom scored.
A flurry of tries then saw Sonatane Takulua and Nili Latu score for Falcons either side of Phil Dollman's effort to leave the game tied at 17-17.
However, late scores from Sam Hill and Elvis Taione eased the Chiefs clear.
Taione's late score ensured the bonus point and temporarily put Exeter two points clear at the top of the table, only for Saracens to then regain the lead with a 26-16 win over London Irish in New Jersey.
Much of the pre-match talk focused on the return of Exeter's England hopeful Henry Slade from injury, but despite some flashes of quality he was unable to spark this game into life.
Instead that mantle fell to Steenson, who was excellent with the boot and probed and sniped to give Exeter momentum.
Rob Baxter's Chiefs have recovered from a mini-wobble earlier in the campaign to stake their Premiership claims.
At the other end of the table, the Falcons remain second from bottom after leaving with nothing to show for their determined display following the late tries from Hill and Taione.
Exeter: Dollman; Woodburn, Slade, Hill, Whitten; Steenson, Chudley; Moon, Yeandle, Williams, Stevenson, Parling, Armand, Salvi, Waldrom.
Replacements: Taione, Hepburn, Rimmer, Atkins, Ewers, Lewis, Hooley, Bodilly.
Newcastle:  Tait; Venditti, Harris, Pablo Socino, Sinoti; Delany, Takulua; Rogers, McGuigan, Cusack, Botha, Green, Wilson, Welch, Latu.
Replacements: Lawson, Harris, Hatakeyama, Mayhew, Hogg, Young, Goode, Penny.
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Premiership title hopefuls Exeter Chiefs left it late to secure a bonus-point win against Newcastle Falcons.