Prior to the discovery of rasagiline, a closely related analog called [https://pubchem.ncbi.nlm.nih.gov/compound/SU-11739 SU-11739] (AGN 1133) was patented. At first, the N-methyl was necessary for the agent to be considered a ring cyclized analog of pargyline with ca. twenty-times the potency. However, the N-methyl compound was a non-selective MAOI.
Racemic rasagiline was discovered and patented by Aspro Nicholas in the 1970s as a drug candidate for treatment of hypertension.
Moussa B. H. Youdim, a biochemist, had been involved in developing selegiline as a drug for Parkinsons, in collaboration with Peter Reiderer.  He wanted to find a similar compound that would have fewer side effects, and around 1977, at about the same time he moved from London to Haifa to join the faculty of Technion, he noticed that rasagiline could potentially be such a compound. He called that compound, AGN 1135.
In 1996 Youdim, in collaboration with scientists from Technion and the US National Institutes of Health, and using compounds developed with Teva Pharmaceuticals, published a paper in which the authors wrote that they were inspired by the racemic nature of deprenyl and the greater activity of one of its stereoisomers, L-deprenyl, which became selegiline,  to explore the qualities of the isomers of the Aspro compound, and they found that the R-isomer had almost all the activity; this is the compound that became rasagiline.  They called the mesylate salt of the R-isomer TVP-1012 and the hydrochloride salt, TVP-101.
Teva and Technion filed patent applications for this racemically pure compound, methods to make it, and methods to use it to treat Parkinsons and other disorders, and Technion eventually assigned its rights to Teva.
Teva began development of rasagiline, and by 1999 was in Phase III trials, and entered into a partnership with Lundbeck in which Lundbeck agreed to share the costs and obtained the joint right to market the drug in Europe.  In 2003 Teva partnered with Eisai, giving Eisai the right to jointly market the drug for Parkinson's in the US, and to co-develop and co-market the drug for Alzheimers and other neurological diseases.
It was approved by the European Medicines Agency for Parkinson's in 2005 and in the US in 2006.