However, United had to ride out an early Southampton storm and the outcome could have been vastly different had Andre Onana not saved Cameron Archer's penalty at 0-0.
De Ligt's first United goal and Rashford's first for six months quickly flipped momentum in the visitors' favour just before half-time.
Southampton's woes were compounded when captain Jack Stephens was sent-off 11 minutes from time for a wild challenge on Garnacho, who rounded off the scoring in stoppage time.
"Sometimes you have to find a way to win," said Ten Hag. "The first minutes we had some problems but after Andre stopped the penalty and the goal by De Ligt I think the game was all ours."
Saints remain without a point from four games on their return to the Premier League, while United's second win of the season lifts them up to eighth.
Ten Hag began the campaign already under scrutiny after surviving an internal review of his position in May following an eighth-place finish in the Premier League last season.
Back-to-back defeats before the international break to Brighton and Liverpool had ramped up the pressure even more on the Dutchman and the Red Devils had to ride their luck early on at St. Mary's.
A failure to make the most of some promising build-up play has been the story of Southampton's their return to the top flight.
Teenager Tyler Dibling caught the eye on his first Premier League start and his trickery lured Diogo Dalot into a rash challenge inside the area.
But Archer's spot-kick lacked conviction and Onana parried his attempt before easily clutching the striker's follow-up header at the second attempt.
'Huge for him'
Just two minutes later, United led as De Ligt put a difficult week with the Dutch national team behind him to open his account.
The former Bayern Munich centre-back was subbed off at half-time by Ronald Koeman after errors that led to two German goals in a 2-2 Nations League draw in Amsterdam.
A short corner caught Southampton napping and Bruno Fernandes' cross perfectly picked out De Ligt to head into the bottom corner.
"Three points from the games is not enough so there was some pressure on this game but we played really well," said De Ligt.
"Big credit to Andre who saved a really important penalty and changed the game for us."
Rashford then ended his wait since March to get on the scoresheet when his low strike from outside the box curled into the far corner on 41 minutes.
"It is so huge for him," added Ten Hag. "For every striker to be on the scoring list is important. Now he has his first, I'm sure there will be more."
Aaron Ramsdale denied De Ligt a second before half-time as United threatened to run riot.
But the second period was far more sedate as the visitors comfortably held Southampton at bay before the home side's frustration bubbled over.
Stephens' unneccessary lunge caught Garnacho on the knee and was rightly punished with a straight red card.
The Argentine then rubbed salt in Southampton wounds with a thumping finish from Dalot's cross in the 96th minute.
