Bruce Arena's Revs had equalised twice, including Tajon Buchanan's 118th-minute leveller, to send the game to a shootout but City goalkeeper Sean Johnson was the hero, denying Adam Buksa's tame effort.
Alexander Callens powered in the decisive penalty to clinch City's maiden Conference finals appearance, ending New England's hopes of a first-ever MLS Cup after setting a single-season record for most points with 73 in 2021.
City will take on Philadelphia Union in the Eastern Conference finals on Sunday, although they will be without top scorer Valentin Castellanos who was sent off after scoring in extra-time.
The visitors went ahead in the third minute with Santiago Rodriguez capping off a fine team move by sweeping home Tayvon Gray's cross.
The Revs levelled in the ninth minute as Polish striker Buksa headed in from Carles Gil's free-kick.
New England's Thomas McNamara headed over the bar in the second half, while Maxi Moralez's cannoned a header into the post for City.
Castellanos blew a great chance late in regulation time, before putting City ahead in extra time by emphatically powering home Gudmundur Thorarinsson's cross.
The Argentinean forward was sent off minutes later for a second bookable offence with a late tackle and the Revs capitalised with Buchanan volleying in Ema Boateng's 118th minute ball.
Buksa almost decided the contest before the shootout, bursting forward but his left-foot shot was pushed over by Johnson.
In the shootout, Johnson denied Buksa again with a low left-hand save and City converted all five of their spotkicks to secure their progress.