The 37-year-old, the only Briton to win a title at Wimbledon in the Open era, had considered retiring in early 2019 due to his struggles with a hip injury but resumed playing after undergoing surgeries.
Playing in front of home fans and with his mother Judy watching, the Briton suffered a blip in the second set when Popyrin served well to force a decider before Murray romped to his 739th win with a volley at the net on match point.
It was also Murray's first win on the Tour since the Miami Open in March as he seeks a sixth title at Queen's despite being a wildcard entry this year.
"I didn't realise, my mum told me it's my 1,000th match on tour. A lot of matches and a lot of wear and tear on the body. It's not easy, but I'm pushing through it," Murray said.
"I haven't gotten too many wins this year. It's been a difficult season, but I did well to come through in the end.
"Any win you can get is important, that's why we're playing. To try and win matches. With Wimbledon a few weeks away, I want to try and build up confidence and get some matches in my legs."
Murray, a two-time champion at Wimbledon, has not moved past the third round at the All England Club in the last three years.
At Queen's he will next face Australia's Jordan Thompson.
Reigning Wimbledon champion and Queen's top seed Carlos Alcaraz breezed into the next round with a 6-1 7-5 victory over Francisco Cerundolo, but second seed Alex de Minaur was upset 1-6 6-4 6-2 by Italy's Lorenzo Musetti.
