Kane's spot-kick against Colombia had England on track for a first knockout win at a major tournament in 12 years, but Yerry Mina's 93rd-minute header sent Tuesday's last-16 tie to extra-time in Moscow.
The England captain subsequently scored from 12 yards again as the match went to penalties, Eric Dier converting the decisive spot-kick after Jordan Pickford saved from Carlos Bacca.
"I'm so proud. We were so unlucky to concede in the last minute. It shows our togetherness and character," Kane told a news conference.
"These are the moments when you really see it. It's a big night for everyone. Obviously we knew that England's history with penalties hasn't been great. It was nice to get that one off our back.
"We have grown up a lot on that pitch tonight. Highs and lows, even in the penalty shoot-out."
Giving England the lead from the penalty spot means Kane has scored in six consecutive games for England, matching a 79-year-old national record held by Tommy Lawton.
"Extremely proud," Kane said of levelling the record. "Hopefully this continues.
"I want to do better and do more. I want to help the team and we're just looking forward to the next game."