The 29-year-old attacking midfielder missed an earlier chance to level for Spurs and acknowledged when he got another chance, he was happy that he was able to come through for his team.
"Yeah it was relief," Dempsey told Spurs TV. "To get a point when you're looking at defeat, it's almost like a win."
"We were trying so hard to get the equalizer," he continued. "It looked like it wasn't going to be our night because the keeper made some great saves during the match. We just knew that getting crosses in was causing them problems. Aaron [Lennon] got a good ball in and [I] put on frame and I'm happy that it went in."
Prior to his late goal, Dempsey would come close to garnering some heavy criticism when he missed an easy shot close to the goal. It appeared like another frustrating moment for Dempsey, who has had a season of highs and lows since joining from Fulham on the transfer deadline.
"Yeah it was difficult. It was one of those things where it was a good ball from Mousa [Dembele] and I got a good touch on it but it was kind of underneath my feet a bit, so I couldn't hit it," Dempsey said. "I tried to hit it hard and low and he [Man United goalkeeper David De Gea] was leaning the other way and made a great save. So yeah, you're thinking that maybe it isn't going to be your day."
He added: "We kept on plugging away, getting shots on goal and we were able to finally get what we deserve."
Dempsey addressed some of his struggles this season, pointing that his form and goal tally is continuing to improve.
"It's been a slow start to the season but now it's good to have five goals in the Premier League and seven in all competitions," Dempsey said. "Hopefully, there are more goals to come because that's why they brought me here. "
Fortunately for Dempsey, his game-winner against United along with his goal from Sunday places him in Spurs history. This is the first season in 23 years that Tottenham has gone undefeated against Manchester United. Dempsey was aware of that fact and stated that those results prove that Spurs are among England's elite clubs.
"Anytime you can get four points against Man United in a season, you've done well," Dempsey said. "We're happy with that return. Any points that we can get at this point of the season are important - especially against the big teams because you want to finish in the top four...
"We're still in good position now, we've just got to keep this run going."
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