Air carrier flight crew reported an NMAC with another aircraft during departure climb.

Narrative

FO was flying; flaps 5 departure from ZZZ XXL. We were cleared for departure with one aircraft on final for rwy XY 5 miles out and one aircraft cleared onto the runway (XXL) behind us. Previously I had heard a different aircraft taxing over for XXL(I believe). We did a rolling takeoff at reduced thrust and lifted off at XA:29. As we were rolling; I heard an aircraft tell tower they would rather runway XY; then tower cleared them to enter downwind for XY. After gear up; I heard tower tell them to pass behind the air carrier. I looked down and saw a TCAS target in very close proximity climbing 800 ft above us. I looked out to the right and couldn't see them. The FO entered a normal turn from rwy heading to 240 when I spotted a single engine Cessna and pointed it out to the FO. She replied that she did not see it. I stated; 'my aircraft' and banked left while stopping the climb and wiping the power; kicking off auto throttles; and keeping the Cessna in sight. I started a slight descent as the Cessna seemed to be turning into (left bank) and descending towards us. We passed about 100-200 feet in front of and slightly below the Cessna. After we were clear; I returned the power to a climb setting and handed the aircraft back to the FO. We notified tower we were visually avoiding the traffic and switched frequencies as assigned. We notified dispatch and called the Chief Pilot upon arrival; along with ZZZ tower to let them know we were filing. It was VFR at the time of the event.