The visitors were two goals down after only four minutes at The Hawthorns following strikes from Matej Vydra and Chris Brunt.
Things went from bad to worse for under-pressure head coach Sherwood when Stephane Sessegnon added a third goal after Emmanuel Adebayor's tame penalty had been saved by Ben Foster.
Tottenham were frail at the back, but looked dangerous going forward and Jonas Olsson's own goal before half-time gave them hope.
Relegation-threatened West Brom surrendered an early two-goal lead to draw 3-3 with Cardiff City in their last home game and capitulated once again as Harry Kane set up a tense finale and then Christian Eriksen denied them three minutes into time added on.
Sherwood, who is widely reported to be on his way out of White Hart Lane at the end of the season, was at a loss to explain his side's defending, but felt they deserved at least a point.
He said: "It was crazy. We might as well put 1-0 up on scoreboard before we start. At 3-0 I thought we were going to win the game. We totally dominated this football match.
"We were disappointed with their third goal, we thought it was a foul, I'm trying to think of as many excuses for our defending, which was horrendous.
"I thought our supporters got us back in the game. They never left us, they are knowledgeable and knew we were battering West Brom.
"They never had a shot on goal in the second half Hugo (Tottenham goalkeeper Lloris) was getting frostbite out there."
The result saw Spurs slip further off the pace in the race for fourth spot, which looks almost certain to be won by Everton or Arsenal.
Sherwood has come under increasing pressure as Tottenham's season has fizzled out, but he insists that he is still enjoying the experience.
He added: "Perversely I am loving it. I like the stick. I am proud to be the manager of this great football club and long may it continue. The fans know I am with them, all I want is the best for this football club."
