Solskjaer had won all five of his games as interim United boss before Sunday's Premier League match at Wembley, seen as the first major test of his reign, AFP reported.
The Norwegian got his tactics spot on in the first half of a match billed as a faceoff between the two favorites for the fulltime job at Old Trafford, with Mauricio Pochettino in the opposition dugout.
Spurs dominated possession in the opening minutes but United's fleet-footed trio of Marcus Rashford, Anthony Martial and Jesse Lingard looked to counter-attack at every opportunity.
The approach paid off just before halftime when the impressive Paul Pogba delivered a raking cross-field ball for Rashford to run onto and the England man drove the ball emphatically past Hugo Lloris, leading to wild scenes of celebration on the bench.
But the second half was a different story as Spurs came forward in relentless waves, only to find De Gea in unbeatable form.
Former United defender Gary Neville hailed the impact his former Old Trafford teammate Solskjaer has had since he replaced the sacked Jose Mourinho last month.
"Ole Gunnar Solskjaer is picking the profile of player Manchester United need and that's players who have the energy to get from box to box quickly and counter-attack with quality," the Sky Sports pundit said on his podcast.
"You've seen that in the last few weeks. At Wembley, it was the acid test. It was the moment where people said could he live in a big game and could he live against the big managers. Second half I thought it was about spirit, desire and hanging on a little bit. The players were playing for the manager."
The hard-fought victory at Wembley certainly boosts Solskjaer's chances of staying at Old Trafford beyond the end of this season and some British bookmakers now make him the favorite ahead of Pochettino.
The Norwegian has done little wrong since he took over an unhappy club from Mourinho, reinvigorating Pogba and Rashford and tightening a leaky defense – the clean sheet against Spurs was just United's fourth of the Premier League campaign.
Solskjaer hailed De Gea as "the best goalkeeper in the world" after the Spaniard's 11 saves and praised his defense but he acknowledged it was tough going after the break.
"Second half, we made hard work of it," he said. "Maybe because of ourselves they created many, many chances but first half I thought it was spot on."
The former United striker will not be happy at how many times the Spurs attack breached his defense and found themselves with only De Gea standing in their way.
Result aside, the Wembley match also showed Pochettino at his best, with his rejigged side cutting through United’s backline at will in the second half.
"After four and a half years, the second half was the best performance I have seen Tottenham play," he said. "It was amazing. In football sometimes you deserve to win and don't and other times, you do not and you win. That is why we love football."
United is now level on points with fifth-placed Arsenal after trailing the Gunners by eight points when Solskjaer replaced Mourinho on December 19 and their bid to qualify for next season's Champions League is gathering pace.
Spurs remain in third place, nine points behind leader Liverpool.
But Neville is warning the United hierarchy not to get carried away by one result.
"It's such a huge win for Ole Gunnar Solskjaer and this group of Manchester United players," he said. "It feels like a million miles away from where this club was a month ago.
"Before the game, I said if Manchester United lost the game, the result should not determine the outcome of the next United manager. One thing I will say is because Manchester won, it shouldn't determine the outcome of the Manchester United manager."