Spurs should have been out of sight by half-time but were frustrated by a series of top-drawer saves by Hart. The England keeper was given the nod over Shay Given and justified his selection with some fine stops as Spurs threatened to run riot, while Gareth Bale beat him but was foiled by the woodwork.
City by contrast played like a team of players who barely knew each other and were given no time on the ball, but managed to get to half-time on level terms.
The visitors improved after the break but it was Spurs who were the more eager for the win. Bale missed an open goal and substitute Roman Pavlyuchenko was denied by a fine challenge from Vincent Kompany.
City escaped with an undeserved point while Spurs may come to rue their missed chances in an entertaining opener to the Premier League season.
Port Vale boss Neil Aspin is expected to choose from an unchanged squad for the home game against Exeter.