Asha Philip, Imani Lansiquot, Ashleigh Nelson and Daryll Neita clocked 42.25 seconds on Friday to come second in their heat behind Jamaica.
Asher-Smith won 200m gold on Wednesday after claiming 100m silver on Sunday.
She will join the team for Saturday's final at the Khalifa International Stadium in Doha.
Philip said: "We're getting ready for the final. Dina and Kat (Johnson-Thompson) have given everyone a boost, two gold medals and it's a good thing for us."
Great Britain won silver in London two years ago to add to the men's world title.
Neita, part of the squad in 2017, added: "The only way is up (from silver), we're coming here with such good preparation."
The men, the defending champions, won their heat in a world leading time this year of 37.56 seconds.
Adam Gemili, Zharnel Hughes, Richard Kilty and Nethaneel Mitchell-Blake impressed in Qatar.
Gemili, who came fourth in the 200m final, said: "I'm actually good. I feel really fresh.
"It's always an honour to come out with these guys. We were looking for redemption, especially myself after Zharnel, after what happened in the individuals."
Kilty added: "We knew the key to this was relaxation. I said to the lads at the start - this is one seamless effort of taking the baton round, and we've done that nicely.
"We're full of confidence, relaxed and ready to go again."
In a surprise the Jamaican squad of Oshane Bailey, Yohan Blake, Rasheed Dwyer and Tyquendo Tracey, failed to make the final.
Earlier, Jake Wightman, Neil Gourley and Josh Kerr all reached Sunday's 1500m final.
Kerr said: "It felt very good. It was a lot better than my heat. I pushed back and felt a lot more confident.
"I knew where I wanted to be. I knew where I wanted to kick from. I'm very confident in how strong and fast I am now.
"We all made mistakes but he got through what it matters and it's great that all three of us are going to be standing on that start line on Sunday, which is fantastic."
The USA's Dalilah Muhammad won the women's 400m hurdles title in a world record time of 52.16secs.
GB's Zak Seddon finished last in the 3,000m steeplechase while home favourite Mutaz Barshim won the high jump with a clearance of 2.37m.
The Bahamas' Steven Gardiner won the 400m crown.
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