Up against Mauricio Pochettino, the favourite to replace Jose Mourinho on a full-time basis at Old Trafford, Solskjaer's team played vibrant football and just about did enough in a match of many chances, with De Gea's display just as important as Marcus Rashford's third goal in as many games.
United was a particular danger on the counter and it was no surprise to see it ultimately go ahead at the end of such of a move, with Rashford finishing after Paul Pogba's excellent pass.
An open second half saw both teams create a raft of chances, but Hugo Lloris and the exceptional De Gea came out on top, with United moving level on 41 points with fifth-placed Arsenal as a result.
After a bright Spurs opening, United looked a threat on the break and went close twice before the 20th minute – Jesse Lingard firing wide of the top-right corner from 12 yards, before Anthony Martial tested Lloris from a tight angle.
Spurs lost Moussa Sissoko to a suspected groin injury after 43 minutes, and United broke the deadlock a few moments later – Pogba's exquisite long ball releasing Rashford, who found the bottom-left corner from 18 yards.
Pochettino's side looked re-energised at the start of the second half, with De Gea making crucial saves from Kane and Dele Alli within five minutes of the restart.
Still United threatened, however, with Pogba seeing a close-range effort tipped over by Lloris, before his controlled effort just inside the box in the 62nd minute forced a smart stop from the France goalkeeper.
De Gea was the star, though, producing a string of saves to keep Kane, Alli and Toby Alderweireld at bay and ensuring Solskjaer equalled Matt Busby's record of winning his first five league games in charge of United.