Arsenal will again play Manchester City in the last four at the Wembley Stadium.
The final is scheduled for Saturday, August 1 at the same venue.
Ings struck in each half to move one goal behind top scorer Jamie Vardy, who is enduring a lean spell for third-place Leicester.
Pep Guardiola's men beat Newcastle 2-0 in their quarter-final, which was played at St James' Park on Sunday.
However, Chelsea's last FA Cup success came at United's expense, as the London outfit won the trophy for the eighth time in their history when they ran out 1-0 winners in 2018.
The much-awaited showdowns will be played at Wembley on the 18th and 19th of July.
It was Sheffield United's first goal since football resumed in England.
At the time of publishing, City were goalless away to Newcastle, who last won the FA Cup in 1955.
There were 1.7m BBC online stream requests for the United victory and 1.1m for City's triumph.
Spanish midfielder Dani Ceballos netted an injury-time victor in a 2-1 win at Sheffield United, which had two goals ruled offside by video replay.
Striker Tammy Abraham had a goal ruled out for offside and then Barkley found space in the area to clip in Willian's measured cross past Kasper Schmeichel.
Harry Maguire struck an extra-time victor as United saw off 10-man Norwich in the first quarter-final on Saturday, while Chelsea beat Leicester 1-0 on Sunday thanks to Ross Barkley's second-half strike.
The win extended United's unbeaten run in all competitions to 14 matches.