Merriam-Webster and Etymonline both connect Persian shakar with Arabic sukkar and the later English word. French sucre lies among the European intermediaries before Middle English sugre.
The sources give differing Italian and Medieval Latin intermediaries, so those are not scored as a fixed chain. The Sanskrit connection is ultimate, not a claim of a direct borrowing at every step.
Sources
- Merriam-Webster: sugar
Word History gives the selected anchors, French mediation and the ultimate Sanskrit source.
- Etymonline: sugar
The main noun entry corroborates the scored anchors and gives a different European chain.
- CNRTL / TLF: sucre
The French entry derives sucre through Italian from Arabic, qualifying the European stages.