The group, which has two sites in the Black Country, also said it was expecting to deliver annual results in line with its expectations.
The cardboard maker, which supplies packaging products to Amazon, Nestle and Unilever, said higher box volumes were offset by a rise in input costs.
Like-for-like corrugated box volume growth had accelerated compared to the second quarter, with e-commerce strength over Christmas continuing into 2021.
"Trading continues to progress well, with the trends and momentum described in our first-half results on December 10 continuing into the second half," the company said.
DS Smith said it also had seen some "encouraging signs of recovery" from its industrial customers.
Rising input costs, together with high demand, continued to drive higher paper prices.
"We have started to recover these additional costs through higher packaging prices and our expectation is, with the customary lag, they will be fully recovered and underpin continued momentum into the 2022 financial year," the company added.
DS Smith has a packaging site at Langley House in Summit Crescent, Smethwick, and a paper and plastics recycling depot at Rose Hill Industrial Estate, Willenhall.
Its major customers include e-commerce and consumer packaged goods companies, which are spending heavily on packaging materials to ensure their products are delivered safely and without any damage.
Miles Roberts, group chief executive, added: “The group has delivered a robust performance during the period against a challenging macro-economic environment, and I remain immensely grateful and proud of our colleagues for their commitment to keeping our plants safe and operational and continued support from our customers.
"We are seeing excellent demand from FMCG and e-commerce customers for our sustainable packaging products and solutions and we continue to invest for growth in these areas. Covid-19 is accelerating a number of the structural growth drivers and with our leading position in recycling and fibre-based packaging we are well positioned to capitalise and drive further market share gains.
"While the economic environment remains uncertain due to Covid-19, we are experiencing good momentum across the business in both Europe and North America. We are confident in delivering results in line with our expectations for the year and showing further good progress and momentum as we move into the next financial year.”