State Street Corp. posted a better-than-expected quarterly profit, as market gains and new business wins lifted revenue.

The custody bank’s shares were up 3% in midafternoon trading.

State Street, whose asset-servicing business performs various accounting and administrative functions for many of the world’s biggest money managers, benefited...

State Street Corp. posted a better-than-expected quarterly profit, as market gains and new business wins lifted revenue.

The custody bank’s shares were up 3% in midafternoon trading.

State Street, whose asset-servicing business performs various accounting and administrative functions for many of the world’s biggest money managers, benefited from a stock-market rally and the flow of new money into its clients’ funds. The bank’s own asset-management business, State Street Global Advisors, also benefited from stock-market gains and inflows into exchange-traded funds.

“It’s been a good year,” Ron O’Hanley, State Street’s chairman and chief executive, said Monday in an interview. “There’s been lots of client activity, and a pretty good market.”

State Street is on pace for its best year ever in winning new business, contributing to an uptick in its assets and servicing fees.

The Boston-based custody bank also struck its first significant deal in three years last month, agreeing to buy Brown Brothers Harriman & Co.’s investor-services unit for $3.5 billion in cash, and ended the quarter with investors speculating about an even bigger transaction. The Wall Street Journal reported last month that the bank was in talks to merge its State Street Global Advisors business with Invesco Ltd.

On the call with analysts, Mr. O’Hanley said owning a money manager was still helpful to both State Street’s servicing arm and its overall financial strength.

“When you add it all together, we like the business,” Mr. O’Hanley said. “But we recognize that you as shareholders can take care of diversification, and it’s not up to us to diversify for you. We have to make sure that as owners of these businesses that we’re good owners, and we can continue to grow the business. We are looking at this all the time.”

State Street’s net income rose 29% to $714 million, or $1.96 a share, in the third quarter. Excluding certain one-time items, the company earned $2 a share. On that basis, analysts polled by FactSet had predicted a $1.92 profit.

Revenue climbed to $2.99 billion from $2.78 billion in the year-ago period. Analysts expected $2.96 billion.

Fee revenue was up 9%, including a 7% rise in servicing fees and a 10% increase in management fees. Net interest income climbed 2% on higher loan balances.

State Street executives said last year the bank would need to bring in about $1.5 trillion in new business each year, all things being equal, to offset attrition and the perpetual pressures on the fees they charge clients and to meet its revenue-growth targets. The bank brought in more than $3 trillion in wins through September, including $1.7 trillion in the third quarter alone, said Eric Aboaf, the company’s finance chief.

State Street ended the quarter with $43.3 trillion in assets under custody, up 18% from a year earlier.

Assets under management climbed 23% to $3.86 trillion, driven by higher market prices and the flow of client money into ETFs. State Street plans to resume its share-repurchase program in the second quarter of 2022 after pausing the program in September.

Write to Justin Baer at justin.baer@wsj.com and Matt Grossman at matt.grossman@wsj.com