Hodgkinson finished in one minute 56.38 seconds, eight hundredths behind the American Olympic champion on the final day of the World Championships in Portland, Oregon.
They went toe-to-toe down the final 100m as Hodgkinson pulled alongside the home favourite but could not pass her.
“I gave it everything right up to the line,” Hodgkinson said.
Hodgkinson said she was “a bit gutted” but added: “A world silver is not too bad. It adds fuel to the fire. I will go away and I still have more championships to come so will have to refocus for that.
“I’m gunning for a bit more so I am looking forward to next year.”
Hodgkinson’s rivalry with Mu, also 20, seems set to define the event for the World Championships in Budapest in 2023, the Olympics in Paris in 2024 and, potentially, many years beyond.
Britain finished with seven medals overall – one gold, one silver and five bronze.
‘I’ve closed the gap’ – Hodgkinson
Hodgkinson and Mu barely broke a smile as they were introduced to the crowd before the start, the tension of the contest clear.
They hit the bell in a swift 57.09 and the race boiled down to the expected face-off down the back straight.
Mu made the first move, stretching her legs to open a lead, but Hodgkinson’s counter followed soon after.
She dragged herself back alongside Mu and seemed poised to pass on the inside, but Mu found just enough to hold on to gold.
As she collected her breath after the finish line, Hodgkinson motioned to her team in the stands, her finger and thumb set a couple of inches apart to show the fine margin of defeat.
“I’ve closed the gap, I’m getting closer,” she said.
“I’ve got a lot of respect for these athletes, particularly Athing. She’s competing in front of a home crowd as Olympic champion, so there was a lot of pressure for a 20-year-old.”
Mu said: “I’m just glad I made it to the line to finish the race, and thankfully I won gold. I just physically wasn’t where I would like to be.”
Relay bronze completes GB medal collection
Hurdler Jessie Knight came into Britain’s 4x400m quartet for the relay final and, together with Victoria Ohuruogu, Nicole Yeargin and Laviai Nielsen, secured a bronze in the 49th and final medal event of the Championships.
A gold-winning United States team were anchored home by Sydney McLaughlin, with Jamaica taking silver.
While Jake Wightman’s 1500m victory was Britain’s solitary gold, Hodgkinson’s silver, along with bronzes for Laura Muir, Dina Asher-Smith and Matthew Hudson-Smith means Britain have five different individual medallists for the first time since 2011.
With Britain’s 4x100m relay team also taking bronze, the haul of seven medals equalled the tally at London 2017 and by a team including a prime Mo Farah and Jessica Ennis-Hill in 2011.
“It has gone much better for the British team than many were expecting,” said former world champion and BBC Sport commentator Steve Cram.
“Earlier this year there was concern around Dina’s form and whether Keely could convert again. Laura Muir was injured as well. In April or May, we were thinking three to five medals would be good.
“It has been a great championships in terms of Britain taking their chances. That is good news heading into an important cycle with Budapest next year and Paris Olympics.”
The US finished top of the medal table with a record 33 medals, including 13 golds. Their tally included podium clean sweeps in the men’s 100m, men’s 200m and men’s shot put.
Ingebrigtsen bounces back from 1500m defeat
Jakob Ingebrigtsen, beaten by Britain’s Jake Wightman in his preferred 1500m earlier in the week, bounced back with a superb victory in the 5000m.
The Norwegian had suggested he would end his season in the immediate aftermath of his defeat by Wightman, but changed his mind and was rewarded with his first world title as he strode clear of Kenya’s Jacob Krop and Uganda’s Oscar Chelimo.
Joshua Cheptegei, who won 5,000m Olympic gold in Tokyo last summer, finished ninth and Britain’s Marc Scott 14th.
Nigeria’s Tobi Amusan, who ran a world record in the 100m hurdles semi-finals earlier in the day, took gold as she sprinted clear of Jamaica’s Britany Anderson and Puerto Rico’s Jasmine Camacho-Quinn.
Amusan’s winning time of 12.06 would have been an improvement on the 12.12 world record she set a couple of hours earlier had it not been ruled ineligible by a tailwind over the legal limit.
Britain’s Cindy Sember was fifth in 12.38, a time that would have further lowered the national record she set in the semi-finals.
Britons Jazmin Sawyers and Lorraine Ugen finished ninth and 10th respectively in the long jump with efforts of 6.62m and 6.53m. Germany’s Malaika Mihambo, who fouled on her first two jumps, defended her world title in 7.12m.