The duo both had to be replaced during last Saturday's match at Manchester City, with March being taken off on a stretcher, as De Zerbi gave an update ahead of Brighton's UEFA Europa League meeting with Ajax on Thursday night.
"Solly has a very tough injury and we will lose him for a long period. Welbeck the same," De Zerbi told the club's official website.
"We have to move on, work harder and find a different solution."
Losing March and Welbeck is a huge blow for De Zerbi, particularly considering their impact in front of goal for the Seagulls.
Since the start of last season, March has provided 18 goal contributions in 40 Premier League matches, while Welbeck has combined for a total of 10 goals and assists in 39 league appearances.
Though losing two more key players is a major setback for Brighton, they still have plenty of options to fill the void.
One player who has impressed since joining the Seagulls this summer is Ansu Fati.
The midfielder, signed on loan from Barcelona, has made five substitute appearances in the league so far, already scoring twice, including in last Saturday's 2-1 defeat to Man City.
De Zerbi also added that there Facundo Buonanotte, Simon Adingra and Evan Ferguson will have more of an influence in the absence of March and Welbeck.
"Facundo Buonanotte will find more space, for sure," De Zerbi added. "[Simon] Adingra will become more important for this team.
"Evan Ferguson will be important, the same as last season."
Those four players could all have the opportunity to impress against Ajax before taking on Fulham on Sunday, with the Seagulls looking to end a run of three league matches without a win.