Djokovic started his quest for an eighth Grand Slam title with a 6-1, 6-2, 6-4 win in the final match of the night at Arthur Ashe Stadium.
The Wimbledon champion Djokovic has a run of four straight finals at the Open, including a championship in 2011 against Rafael Nadal. Djokovic lost last year to Nadal, but the Spaniard isn't defending his title this year due to a right wrist injury.
Playing his first career tour-level main draw match on hard courts, the 22- year-old Schwartzman was no match for the world's No. 1 player. The Serbian breezed through the first two sets in 56 minutes. Schwartzman won the first game of the third set and then had a chance to break, but a smiling Djokovic roared back and evened the set with an ace.
There were a couple of hiccups from Djokovic, as Schwartzman broke to tie it at 3-3, but he couldn't follow it up. Djokovic broke back and then held at love before closing it out.
"It's never easy to start a U.S. Open smoothly," said Djokovic, who had 27 unforced errors, three more than Schwartzman.
Up next for Djokovic will be Frenchman Paul-Henri Mathieu, who topped Gilles Muller of Luxembourg in a five-set marathon.
"I'm sure we'll both be fresh and motivated to play," Djokovic said. "He's a very solid player from the baseline, but I'll focus on myself and hope to do as well as I did tonight."
Murray, the eighth seed and former world No. 2 star, battled cramps before holding off wily Dutchman Robin Haase 6-3, 7-6 (8-6), 1-6, 7-5 in 3 hours, 8 minutes.
"I don't know what it was. I was cramping after 1 hour and 45 minutes," Murray said. "I didn't know what to do in the third set. I started to get it in my laterals and forearms when I was serving. I didn't know whether to conserve energy or try to finish it.
"I tried to hang around and tried to play without using my legs much. I managed to get through. These slams are physically challenging, but I need to work out why it happened. It shouldn't have happened, regardless of the temperature."
Murray overcame a 1-4 deficit in the fourth set to get off the court before a fifth.
The two-time Grand Slam champion was the U.S. Open winner two years ago and runner-up in Flushing in 2008. His second-round opponent will be German Matthias Bachinger.
Wawrinka, this year's Australian Open champion, topped Czech Jiri Vesely, 6-2, 7-6 (8-6), 7-6 (7-3). Wawrinka, who is from Switzerland, reached his first U.S. Open semifinal last year before losing to Djokovic.
The Canadian Raonic eased past Japan's Taro Daniel, 6-3, 6-2, 7-6 (7-1). Raonic, who has reached the fourth round here the last two years, is trying to build on a semifinal appearance at Wimbledon when he lost to Roger Federer.
In other opening-day action involving seeds, No. 16 Tommy Robredo of Spain defeated Frenchman Edouard Roger-Vasselin, 6-4, 6-3, 6-4; promising Aussie Nick Kyrgios upended No. 21 Russian and two-time U.S. Open semifinalist Mikhail Youzhny 7-5, 7-6 (7-4), 2-6, 7-6 (7-1); No. 22 German Philipp Kohlschreiber handled Argentine Facundo Bagnis 6-2, 7-6 (7-3), 6-3; No. 23 Argentine Leonardo Mayer was leading Albert Montanes 6-2, 3-0 when the Spaniard retired; Benoit Paire took out No. 24 fellow Frenchman Julien Benneteau 7-6 (7-4), 5-7, 6-4, 4-6, 6-4; 30th seed Jeremy Chardy of France toppled Colombian Alejandro Falla, 6-7 (5-7), 6-2, 7-5, 6-4; and No. 31 Spaniard Fernando Verdasco outlasted Slovenian Blaz Rola 6-3, 3-6, 7-5, 1-6, 6-4.
Several other men moved on, including Italian players Andreas Seppi and Simone Bolelli.
Federer, the second seed and five-time U.S. Open champion, plays his first- round match Tuesday night against Australian Marinko Matosevic. Federer hasn't won the U.S. Open since 2008, the last of his five consecutive championships at the event.
Spain's David Ferrer, the No. 4 seed, faces Damir Dzumhur of Bosnia and Herzegovina during the day session at Louis Armstrong Stadium. Other seeds to see action on Tuesday include, No. 10 seed Kei Nishikori, 12th seed Richard Gasquet and 13th seed John Isner, who takes on fellow American Marcos Giron.