Manager Andre Villas-Boas has come under pressure after his side slumped to an embarrassing 6-0 defeat at the Etihad Stadium last weekend.
That heavy reverse capped off a month that saw Tottenham fail to register a Premier League goal throughout November, but December brought a change in fortunes as Walker and Sandro found the target against the champions on Sunday.
The impressive Wayne Rooney twice equalised to deny the hosts victory at White Hart Lane, but Walker was encouraged by Tottenham's display.
"We needed to show the fans something and prove to ourselves we're a top team. We've stayed strong this week," the full-back told BT Sport.
"The results haven't been great but we've stayed strong as a team. The spirit in the changing room is first-class."
Walker is confident Tottenham, who remained ninth as a result of Sunday's draw, can finish in the top four.
"We've brought a lot of players in and it might take a bit of time. The players we have got are top-class international players so we'll be up there. I guarantee it," he added.
Villas-Boas was invited to send a message to any frustrated Tottenham fans, but declined.
The Portuguese expressed satisfaction with his team's response to last weekend's defeat, however, and felt three points would have represented a just reward.
"To respond in this fashion against a team that has just put five past the second team in Germany (a reference to United’s 5-0 UEFA Champions League win over Bayer Leverkusen in midweek) was good for us," said the Tottenham boss.
"We are extremely happy with the performance, not so much with the result. I think we probably deserved to win the game."
