Spurs fell 2-0 behind on aggregate when Roman Zozulya headed the visitors ahead at White Hart Lane at the start of the second half before goals from Christian Eriksen and a brace from Emmanuel Adebayor, with Zozulya sent off in-between, sealing a memorable 3-2 aggregate win for Sherwood and his players.
They will now meet Benfica in the last 16 and Sherwood was left hailing his players for how they reacted when their backs were against the wall.
"I was delighted with the character," he said.
"I have got a liberty to question their character when they play like they did tonight.
"After conceding so early in the second half it was a real blow for us but it was all based on character and desire.
"I always felt we needed to score three goals to go through tonight.
"It was a bit of a nothing game in the first half but we opened them up time and again in the second half."
Sherwood plans to squirrel away the feelings of jubilation from the comeback to inspire his squad should they hit a sticky patch before the end of the season. Spurs will chase Europa League success and a top-four finish in the Barclays Premier League over the coming months.
"You can only learn from experiences," he said.
"This will be a good example of that, many of you probably thought we were dead and buried but the lads really showed belief.
"It was electric out there after and it was only us who were going to keep scoring.
"It is a good platform, a good learning curve and something we can point to in the future."
Toronto-bound Jermain Defoe said his final farewells to White Hart Lane at the interval and Sherwood admitted he was keen for the Tottenham favourite to get the crowd behind his current crop.
"I was saying to Jermain 'have you been out on the pitch yet?' He told me he had but I told him to get out there again to whip up a frenzy.
"He deserves it, he is a legend, a credit to the football club and the English game."
Sherwood felt Zozulya was fully deserving of his dismissal after headbutting Jan Vertonghen - who appeared to fall easily after contact was made - but said he did not see Roberto Soldado elbow on Artem Fedetskiy.
"Did he? I haven't seen it," Sherwood said when asked about Soldado's indiscretion.
"I think the guy (Zozulya) sticks his nut in his (Vertonghen's) nose so you have got no chance of staying on the pitch when you do that have you?
"I didn't see any decisions which were too out of the way, I thought he refereed the game well. I think Dnipro played a good game and are a good side, they may go on and win that division."
Adebayor struck his brace a day after his 30th birthday and was quick to thank Sherwood for giving him a run in the first team after he suffered a spell on the sidelines under former boss Andre Villas-Boas.
"I'm very glad at being on the pitch, scoring goals, playing very well for my team, helping my team and I just have to keep the focus, keep going, keep playing well," he told ITV Sport.
"I want to say big thanks for the new gaffer, who came in and gave me my confidence back, gave me the stage to play on.
"Everything's going well for me, for him and for the club.
"We just have to keep focused and keep going to the end of the season."
Dnipro, managed by former Tottenham boss Juande Ramos, put up another good showing after their win in Ukraine last week, and the Spaniard was in a reflective mood.
"It was a pity that such a beautiful game was affected by the controversial decisions from the referee," he said.
"We played a good game and controlled the ball with confidence. We scored and then I believe the referee was strict with the red card and that inspired the Tottenham side."