It was their second successive ‘Old Firm’ loss after going down 1-0 away to Rangers B the previous week.
Celtic scored both their goals in the first half with Moffat giving them the lead after 20 minutes when he reacted first to pounce on a shot parried by home goalkeeper McNeil.
Striker Carse doubled their advantage after 33 minutes when the ball fell into his path and he finished well.
The defeat leaves Berwick tenth in the table with 25 points from 18 games played.
On Tuesday Berwick are home to Edinburgh University in the league.
In the East of Scotland League, Tweedmouth Rangers had a welcome 2-1 home win over third in the table Kennoway Star Hearts.
The win lifts then to 12th in the table with nine points (three wins) from their 14 games.
On Saturday they are home to Ormiston, who are joint bottom of the table on seven points.
In the Border Amateur League, Tweedmouth Amateurs maintained their unbeaten run at the top of the B Division with a 2-1 away win over Kelso Thistle. Goals for the Ammies came from Jack Sprot and Aaron Hope.
Spittal Rovers lost 4-2 at home to Stewarton United in the third round of the Scottish Amateur Cup.
The Berwick side led at half time courtesy of a double from Grey, but the visitors turned the match in their favour in the second half with four goals of their own.
In the C Division, Highfields United suffered their first defeat in five games wh3n the6 went down 4-3 at home to Eyemouth United.
BAL fixtures for Saturday are:
B Division - Tweedmouth Ams v Selkirk Victoria; C Division - Gala Fairydean Rovers v Spittal Rovers; Hawick Colts v Highfields United; St Boswell’s v Berwick Colts.
In the Northern Alliance, North Sunderland won 2-1 away from home against Red House Farm in the Cowey Cup. Goalscorers for the Seahouses side were Ross Moore and Tom Allan.
On Saturday NS are home to Ellington Development in the C Division.
In the North Northumberland League, Wooler lost 3-2 away to Alnwick A.