Kapoor will take over on 1 September. Becht will stay on as a part-time adviser until September 2012 to ensure a smooth transition.
Kapoor has been with RB for 25 years and has held a number of key global marketing roles.
RB's brands include Cillit Bang, Air Wick and Dettol.
Becht is retiring at the age of 54. In 2009, he became the highest paid chief executive in the FTSE 100 when he took home £92m, after he exercised long-held share options worth £74m.
But he was praised for donating £110m of shares to a charitable trust, which supports charities like Save the Children.
In 2010, Becht earned a total of £18.2m, made up of £4.2m in basic pay, along with performance-based payments, benefits and pension contributions.
Kapoor joined RB in 1987, when he worked at the company’s Indian operations.
His marketing roles with RB have included vice-president marketing India and regional South Asia for RB's earlier incarnation Reckitt & Colman and UK senior vice president global category officer home care.
Becht has been chief executive at RB since it was created in 1999 through the merger of Benckiser with Reckitt & Colman.
The Dutchman joined the Benckiser side of the business in 1988, after a career at Procter & Gamble.
During his tenure, RB has been one of the top 10 UK advertisers, ploughing money behind cleaning and healthcare products like Vanish, Nurofen and Strepsils.
Key acquisitions include the purchase of Boots’ over-the-counter (OTC) business, Boots International, including Nurofen, for £1.9bn in 2005.
In 2008, RB acquired Adams Respiratory Therapeutics for $2.3bn (£1.4bn).
It bought SSL International, the maker of Durex, in 2010 for £2.5bn.
According to its latest financial results, RB reported revenues of £8.45bn in 2010, up 9% on 2009. Pre-tax profits grew 13% to £2.13bn.