Mauricio Pochettino's side earned a draw against Pep Guardiola's domestic treble winners over the weekend, despite facing an onslaught of pressure from the first whistle to the last.
City twice took the lead in the match, through Raheem Sterling and Sergio Aguero, but Spurs pegged them back on both occasions via an Erik Lamela strike and a Lucas Moura header.
Gabriel Jesus thought he had snatched all three points for City in added time, but his effort was ruled out by VAR after Aymeric Laporte had handled the ball in the build-up.
Guardiola's men had 30 shots in total during the match to Tottenham's six, but the two sides ultimately had to settle for a point apiece in their second fixture of the new Premier League season.
Winks, who saw 56 minutes of action at the Etihad, insists that Spurs have proved that they can "compete with the very best" and he is hopeful that the team can remain consistent as the campaign progresses.
"I think they (City and Liverpool) should worry about us," the midfielder told the Evening Standard. "We've shown in the last few years we can compete with the best teams in the Premier League and Europe. It's important we do that on a consistent basis this season and it's important we go to places like City and Liverpool and get results.
"The manager doesn't allow fear in us. We don't go into any game beaten mentally. We go into every game knowing we can win and we can outperform them. Our mentality going into every game is to be on the front foot, press and try to win. That's probably why we came away with a good point."
Winks also confessed that Tottenham's display against City was "messy" but feels Pochettino got his tactics spot on against an opponent renowned for dominating possession.
"They're an excellent side, they're champions and you have to respect them but like all sides they have weaknesses and I think we exploited them, especially from the corner," he added.
"We know how good we are. We know we're a top side but when we come to places like this it's important to get a result.
"At times it was messy, it was difficult but that's what you have to do, especially at City. You expect City to have a lot of possession, which they did. But it's important when you come here to weather the storm when you're under the cosh, to stay in there and I thought we did that fantastically."
Next up for Spurs is a home clash on Sunday against Newcastle, who have kicked off their latest campaign with defeats against Arsenal and Norwich.