A year ago to the day, United lost 2-0 at Arsenal to fall 24 points behind the English champions, having played two games more.
Another season contending just for a top-four finish looked in store for Solskjaer’s men after winning only two of their opening six league games.
But as all the other contenders have stumbled in recent weeks, the Red Devils have now taken 26 from a possible 30 points to mount an unexpected title challenge.
“Of course we are happy with what we are doing. We have shown we have improved a lot in a year,” said Solskjaer.
“We are getting fitter and fitter and stronger and stronger.”
Bruno Fernandes has been the key figure in transforming United’s fortunes over the past 12 months and the Portuguese again provided the winner to edge out a rapidly improving Villa side.
Solskjaer’s side needed a 93rd minute winner to beat Wolves on Tuesday, but were far faster out the blocks this time round.
Anthony Martial forced Emiliano Martinez into an excellent save high to his left inside 10 minutes before Fred, Paul Pogba and Fernandes fired efforts just off target.
The breakthrough came five minutes before the break and owed much to Pogba’s ingenuity with the Frenchman finally starting to find form despite the speculation over his future.
Pogba and Marcus Rashford combined to send Aaron Wan-Bissaka free down the right and his cross was headed home by Martial at the near post.
Villa were on a five-game unbeaten run themselves to earn a place in the top six and showed why with their start to the second-half as United were forced back.
Jack Grealish was again at the heart of all the visitors’ best work going forward and it was from his cross that Bertrand Traore equalised.
However, Dean Smith’s men quickly undid their good work as a soft challenge from Douglas Luiz on Pogba was deemed enough for a penalty by referee Michael Oliver.
Fernandes’s near-perfect penalty record continued as he slotted the spot-kick just out of the reach of Martinez for his 15th goal of the season.
United needed a big save from David De Gea to deny Matty Cash and a brave block from Eric Baily in stoppage time to hang on and Solskjaer is not getting carried away with talk of challenging for the title ahead of a trip to Liverpool in two weeks’ time.
“It’s early on in the season and we shouldn’t be carried away,” added the Norwegian. “The league this season is going to be so tight.
“There are going to be difficult games, strange results, but since ‘Project Restart’ I think we’ve been very consistent.”
Everton lack ‘quality’
Earlier, Everton missed the chance to close to within a point of the top two as they were beaten 1-0 at home by West Ham.
The Toffees failed to impose themselves on a West Ham side playing their third game in six days and were punished when Tomas Soucek slotted home the winner four minutes from time.
“It was not a good game,” said Everton boss Carlo Ancelotti.
“We didn’t have opportunities and we didn’t concede opportunities, they had a lucky deflection and we lost the game. We could play with more intensity and more quality up front.”
David Moyes’s men had the toughest festive fixture schedule in the Premier League in terms of rest and he said there has been a change in mindset with the Hammers this season to battle through adversity.
“The players have played three games in six games, they are amazing,” said Moyes, who won for the first time as a visiting manager at Goodison Park, where he made his name in 11 years in charge of Everton.
“I don’t think there was any difference in the fitness levels so great credit to the players, there has been a big change in the mentality here at West Ham.”
Victory took West Ham to within three points of third, although they remain in 10th in a tight Premier League table.