Earlier, three-time league winners Saracens, who beat Clermont Auvergne to retain their European Champions Cup title last week, appeared to have booked a place in the Twickenham final when Mike Ellery scored a 75th-minute try at Sandy Park.
However, Simmonds settled the match at the death following a superb passage of play as Exeter avenged last season's final defeat by Sarries and denied the champions the chance to repeat last year's European-Premiership double.
"We had the game with 90 seconds to go but that's the game isn't it?" Saracens Director of Rugby Mark McCall told the club's official website. "But we can unbelievably proud.
"If you're going to lose, lose like that," he added. "We're naturally sad with the result but we fought for everything."
London club Saracens showed no signs of fatigue after their continental exploits at the weekend and took a 6-0 lead after 15 minutes following two penalty goals by Owen Farrell.
However, the hosts responded before the break as two Gareth Steenson penalties levelled the scores following a spell of intense pressure from Chiefs.
Exeter, seeking a first title, picked up where they left off after halftime, with Jack Nowell scoring the first try of the match after picking up from a ruck, before Steenson converted to put them 13-6 ahead.
Saracens held their nerve to remain within touching distance despite the hosts' dominance and reduced the deficit in the 56th minute when Chris Wyles scored in the corner, although Farrell missed the conversion.
During a tense final 10 minutes, Saracens' characteristic composure deserted them as Chiefs remained resolute and looked set to see out victory until the visitors' Ellery barrelled past Henry Slade after a magnificent pass from Schalk Brits.
A superb penalty kick to touch from Slade gave the hosts one final chance, and Simmonds drove over from five metres out after a trademark maul in the 79th minute as Exeter extended their unbeaten run in 2017 to 11 league games.
At the Ricoh Arena in Coventry, Wasps, who ended the regular season in top spot, established a 10-point lead inside the first 10 minutes with two Jimmy Gopperth kicks either side of Kurtley Beale's try.
Tigers reduced the arrears with two fine penalty kicks from Freddie Burns and a Peter Betham try, but another Gopperth kick before the break stretched Wasps' lead.
The visitors heaped the pressure on to break the hosts' resolve in the 52nd minute when Telusa Veainu went over out wide and Burns converted before the dramatic finale which saw Bassett squeeze over in the corner two minutes from time.
(Reporting by Ed Dove; Editing by Ken Ferris and Ian Chadband)
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