B737-700
Flight Crew Air Transport Pilot (ATP); Flight Crew Multiengine; Flight Crew Instrument
Confusion; Human-Machine Interface
Deviation / Discrepancy - Procedural Published Material / Policy; Inflight Event / Encounter CFTT / CFIT
Air Traffic Control Issued Advisory / Alert; Flight Crew Became Reoriented; Flight Crew Returned To Clearance
Human Factors
We were on a visual approach to Runway XXL at ZZZ at 3;000 feet. Approach cleared us for a visual approach about 7 miles out and we were giving a heading towards the FAF following an airbus. PF set 2;000 feet in the MCP and we began a slow descent heading inbound to the airport. ATC issued a 'low altitude alert' at about 2;300 feet. We leveled off and intercepted the G/S just past the FAF. In addition; we had to rapidly slow since the airbus in front slowed rapidly and ATC wanted us to slow to approximately 150 knots or less. They really did not leave us much lateral room. We slowed and configured while descending. We were never near an obstacle; nor were we a threat to another aircraft. We were still above the 1;000 feet stable approach criteria (1;300 to 1;500 feet) and met the criteria proof to reaching 1;000 feet. We were lower than what was normally flow for that area but still seamed safe. Again on a visual approach. PF stated that he read 2;000 feet at the FAF and misread the altitude. The FAF altitude is 2;800 feet. I did not reverify it since we were visual and I was working on setting up radios and configuration for landing. I assumed he saw it correctly.I can't find where we violated anything. We were a little low for the area but still in a safe position to land and had visual clearance from all obstacles. I know next time to verify the altitude in the MCP from the chart myself. That is that only thing that may have stopped the lower alt in the first place.
On receiving landing clearance; I set 2;000 feet in the altitude window and planned to descend to that altitude at ZZZZZ; the FAF.  I misread the approach plate. The correct altitude at ZZZZZ is 2;800 feet. The Tower gave us a Low Altitude Alert at 2;200 MSL in VMC and we maintained that altitude (we were nearly at glideslope intercept) until glideslope intercept.  We landed uneventfully.
Air carrier flight crew reported entering the wrong altitude in the MCP and descending below FAF altitude which resulted in Air Traffic Control issuing a low altitude alert.