Kaspi was born in Austin, Texas, but her family moved to Canada when she was seven years old. She completed her undergraduate studies at McGill in 1989, and went to Princeton University for her graduate studies, completing her Ph.D. in 1993 under the supervision of Nobel Prize-winning astrophysicist Joseph Taylor After positions at the California Institute of Technology, the Jet Propulsion Laboratory, and the Massachusetts Institute of Technology, she took a faculty position at McGill in 1999. At McGill, she held one of McGill's first Canada Research Chairs, and in 2006 she was named the Lorne Trottier Professor of Astrophysics. She is also a Fellow in the Canadian Institute for Advanced Research.
Kaspi's husband, David Langleben, is a cardiologist at McGill and the chief of cardiology at Sir Mortimer B. Davis Jewish General Hospital in Montreal.
Kaspi's observations of the pulsar associated with supernova remnant G11.2−0.3 in the constellation Sagittarius, using the Chandra X-ray Observatory, showed that the pulsar was at the precise center of the supernova, which had been observed in 386 CE by the Chinese. This pulsar was only the second known pulsar to be associated with a supernova remnant, the first being the one in the Crab Nebula, and her studies greatly strengthened the conjectured relationship between pulsars and supernovae. Additionally, this observation cast into doubt previous methods of dating pulsars by their spin rate; these methods gave the pulsar an age that was 12 times too high to match the supernova.
Kaspi's research with the Rossi X-ray Timing Explorer showed that soft gamma repeaters, astronomical sources of irregular gamma ray bursts, and anomalous X-ray pulsars, slowly rotating pulsars with high magnetic fields, could both be explained as magnetars.
She also helped discover the pulsar with the fastest known rotation rate, PSR J1748-2446ad, star clusters with a high concentration of pulsars, and (using the Green Bank Telescope) the "cosmic recycling" of a slow-spinning pulsar into a much faster millisecond pulsar.
Kaspi won the Annie J. Cannon Award in Astronomy of the American Astronomical Society in 1998, the Herzberg Medal of the Canadian Association of Physicists in 2004, the Steacie Prize in 2006, the Rutherford Memorial Medal of the Royal Society of Canada in 2007, and the Prix Marie-Victorin, the highest scientific award of the province of Québec, in 2009. In 2010, she was made a Fellow of the Royal Society. Kaspi received the John C. Polanyi Award in 2010. In 2016, she was awarded the Gerhard Herzberg Canada Gold Medal for Science and Engineering, the first woman to receive this prize. In 2016, Kaspi was appointed as a companion of the Order of Canada, Canada's second highest civilian honour.