It didn't start well for Sir Alex Ferguson's men when Celtic striker Jan Vennegoor of Hesselink unlocked the United defence to open the scoring after 21 minutes.
Saha's double then put the home side ahead before a dipping free kick from Japanese midfielder Shunsuke Nakamura left goalkeeper Edwin van der Sar stranded as the visiting fans celebrated.
But just two minutes into the second half crowd favourite Solskjaer scored what was to prove the winner as he netted United's third goal.
After the match Saha was keen to distance himself from comparisons of departed Dutch striker Ruud van Nistelrooy.
"It was very important to get off to a winning start after last season's problems in Europe and that's the main thing for me, not the goals I scored," the former Fulham striker insisted.
"It was great to score twice, but I know that I maybe should have scored more goals. I could have scored a hat-trick, but at least we got the three points.
"I know people are talking about me replacing Ruud, but I just want to give 100 percent and do well in each game. I just want to keep my place by doing well."
United's three goals means they have already equalled their tally from last season's disastrous campaign.
The other Group F match was not nearly as exciting as FC Copenhagen and Benfica drew 0-0.
Arsenal break duck
They were even thinking of Manchester United in Hamburg as Arsenal manager Arsene Wenger quickly turned his sights to Sunday's domestic showdown with their arch rivals after defeating a 10-man Hamburg side 2-1 in Germany.
Hamburg were reduced to ten players after just 12 minutes with goalkeeper Sascha Kirschstein sent off for felling Robin van Persie in the penalty area.
Brazilian international Gilberto Silva converted the resulting spot-kick to put Arsenal ahead and the visitors were in cruise control.
Robin van Persie helps reduce
Hamburg to 10 men
Czech Republic captain Tomas Rosicky enjoyed a happy homecoming to Germany as he scored a stunning strike to clinch all three points.
"It is nice to score so early on in my Arsenal career and to get off with a win," said former the former Borussia Dortmund midfielder.
"It is our target to win this competition but there are a lot of good teams in this competition. We are just one of them."
Manager Wenger was immediately focused back on the challenge at home.
"It was good to get players back and get them physically fit and sharp. It was important to win so we can go into the Manchester United match knowing how to win as we have not managed to do it in the league," Wenger said.
"I think you will see an interesting game on Sunday. Both teams are always up for it and it is always an intriguing game when the two teams meet."
The points were shared in Portugal as Porto and CSKA Moscow drew 0-0 to send Hamburg to the bottom of Group G.
Lyon lash Madrid
In Group E, it was another ominous night for Real Madrid after they were outclassed by Lyon 2-0.
The scoreline could have been much worse had it not been for the efforts of Madrid keeper Iker Casillas, but in the end the French champions were satisfied with two first half strikes from Fred and Tiago sealing their victory.
Despite Lyon's victory Romania's Steaua Bucharest claimed top spot with a 4-1 hammering of Dynamo Kiev, while in Group H AC Milan began their campaign with a 3-0 win over AEK Athens as Anderlecht and Lille shared the points.