Despite the late elation sparked by Dominic Solanke's 92nd-minute equaliser, Tottenham remain in the relegation zone with just three points from their opening six matches.
It is just the fourth time the club have failed to win any of their opening six games of a league season, after 2008-09, 1955-56 and 1912-13.
Spurs have often looked lively in attack this season, but they have lacked conviction in front of goal and only 19th-placed Coventry have scored fewer goals than Tottenham's three.
However, Tottenham fans can take belief from their overall performance at Old Trafford as they ran Manchester United ragged in the opening half with fluid, high-tempo football.
"For the first time this season we showed we are a team," boss Roberto de Zerbi said.
"We played for the team-mates, we played together, we stayed inside of the game in every situation and we attacked as a team and defended as a team."
The Italian manager added on Sky Sports: "I'm very happy because I've seen the right spirit, the right character and I want to start from this character, this reaction, this behaviour."
Tottenham spent more than £250m on new talent over the summer - adding 10 players to their squad.
But their poor start has prompted the club to organise a warm-weather training camp in Spain in the hope it will provide a chance for their new-look squad to bond.
"Now the target is to build the team," De Zerbi said.
"We are a great group of players but we are not a team yet; to be a team is to play for the team-mates, suffer together, attack together."