B737 flight crew reported on final approach went below the glideslope and ATC advised them of their low altitude. They disconnected the autopilot and climbed to intercept the glideslope.

Narrative

Prior to this red eye flight I was able to get a 2-3 hour nap in. Most of the flight went on uneventful. While getting vectors to final for Runway XXR we were advised that we would be following a heavy aircraft on approach. We were told to proceed direct to ZZZZZ. Just prior to crossing ZZZZZ we were told to expect vectors across final for traffic; but I had to inquire what heading they wanted as we were presently in LNAV; already in the turn to the final approach course. By the time they had us turn back towards the runway and gave us a lower altitude; we were a little higher and further off course than I would have expected. The vectors they gave us pointed us toward the departure of the runway rather than back towards the final approach fix. They asked us if we saw the heavy traffic on final and cleared us for the visual. We set in 1600 (FAF altitude) and the Captain turned the heading bug a little farther to give us a better intercept angle while we continued to slow down and configure the aircraft. We realized that the heading we had set would not allow us to intercept the localizer before crossing the FAF and we captured 1600 so the Captain called for TDZE to ensure we didn't level off and end up high on the approach. When I set this altitude; the pitch mode went to VS -750; which seemed like a proper descent rate to me based on where we were along the approach. What I didn't realize was that the Captain then switched to LVL change; which caused us to descend much faster at idle power. While I was focused on our course and finishing configuring the aircraft and trying to get a word in with Tower (I was waiting for a break in the radio chatter to make my first Tower call); ATC called us about us being low and to check our altitude. Right around that same time I noticed that the below G/S switch light was illuminated and I could just barely hear an oral cue going off (it was so quiet that I couldn't really hear what it was saying). I then looked at our altitude and quickly outside and stated 'altitude' to alert the captain that we were too low and needed to climb. He then turned off the autopilot and initiated a climb back up towards 1000ft to reintercept the glidelope. We did get below glideslope once more on very short final but we were visual and hand flying and were indicating 3 red; 1 white on the papi. We landed normally and taxied to the gate.