Solskjaer took over from Jose Mourinho in December and since his arrival, the Red Devils have won eight of the nine games.
United now sit in sixth position and are five points behind Chelsea in fourth, but they do have a game in hand on the Blues.
With 14 games left, Solskjaer is confident his team can get the job done.
United’s next game is against Leicester on Sunday in the Premier League.
“I feel it’s very [well] equipped. We’ve had now six weeks together. We’ve found out what strengths, weaknesses we’ve got.
“I think we’ve risen to most of the challenges, maybe of course the setback of the two Burnley goals we conceded.
“Maybe they will make us move to the next level because I think we’ve given ourselves a chance now to get into the top four maybe earlier than we thought.
“It’s not too many games, there’s still a week between every single one of them but it’s tough opposition. Definitely.
“I’ve got a big squad, it’s only Marcos Rojo now who’s out for a period, the rest are fit and available,” Solskjaer told Sky Sports.