But Milan Skriniar turned the ball into his own net after 87 minutes with Gonzalo Higuain heading in the victor two minutes later.
Juve lead Serie A by one point after losing 1-0 to second-placed Napoli last Sunday and defeat by Inter will give their rivals a chance to go top.
"The match against Juventus will be very tough but we're ready to play for our objective of Champions League football in this final part of the season and I wouldn't have any issue with going to Lazio on the final day to fight for this goal". Minutes later Vecino was shown red for a foul on Mario Mandzukic. Mauro Icardi equalised shortly after half-time and the veteran Juventus defender Andrea Barzagli turned Ivan Perisic's low cross into his own net.
Juve were facing a dramatic defeat until Cuadrado's shot from an impossible angle deflected in off Skriniar with three minutes left.
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Juve coach Massimiliano Allegri was sent off for entering the pitch after Higuain's goal.
Earlier 10-man Roma warmed-up for next week's must-win Champions League clash with Liverpool by easing to a 4-1 win over struggling Chievo to consolidate third place.
"NO ONE is more devastated about the result tonight than Davide". The Tuscan club are clearly concerned about ramifications if away supporters end up mixing in with home fans, and although sales will not be allowed to Campania residents, there is clearly a potential for trouble in this high-stakes match.
Inter Milan had a rough night last night as they lost to Juventus. "I'm interested in what the three points mean for us, not denying Juventus three points. And how must we be after seven years without winning anything?"