But tries from Quins scrum-half Sam Hidalgo-Clyne and wing Charlie Walker, with Smith converting one, against an unconverted score from prop Harry Williams was just not enough for the win.
Exeter began like an express train, scoring two tries in the opening 11 minutes as Quins struggled to get a touch of the ball.
Devoto got the first as flanker Dave Ewers broke away in midfield on the Quins 10-metre line and, with no defence at home, he gave it to his colleague to fly over unopposed from 22 metres.
Simmonds added the extras and, after an intense passage of possession stretching from the halfway line to the Quins posts, White to pick up from a ruck a metre out to tap down.
It was impressive from the Chiefs but the Londoners, desperate to get something from the game in their play-offs bid, turned the tables.
Number eight Alex Dombrandt was put away down the touchline by centre Francis Saili was over in the corner only to be brought back for a foot out of play.
For all their territorial advantage, however, Quins had nothing to show for it until the home side infringed at a scrum to give Smith a simple penalty on 40 minutes.
Exeter hit Quins with another sucker punch at the start of the second half when a tiny gap appeared that replacement Sam Hill exploited. He was tackled inside the opposition 22, but his team-mates continued the momentum for Williams to score at a ruck seconds later.
Directly from the restart, Simmonds dropped the kick-off, replacement hooker Elia Elia kicked on but White won the race as the ball crossed the home line.
Yet again, captain Chris Robshaw lead Harlequins to pressurise Exeter yet they could not unlock a stubborn home defence despite the Chiefs losing replacement hooker Ben Moon to the sin-bin for a scrum offence just after the hour.
Finally, their efforts were rewarded as a ruck under the Exeter posts saw Hidalgo-Clyne touch the ball against the base of the post. The conversion was simple for Smith.
And once that breach had been made, Quins got another in quick time from a smart move which saw Hidalgo-Clyne and Smith combine to put Walker over in the corner. Smith's conversion to tie the scores went wide.
Smith missed a 35-metre penalty five minutes from time and Exeter clung on for victory.