On the clay in Casablanca, Anderson had no answers for the Romanian, who wrapped up a 6-4 6-4 triumph inside 90 minutes at the ATP 250 event.
Last year's runner-up was broken in the fifth game of the first set as Hanescu claimed the early advantage - and although he squandered two set points in the ninth game - he held in the next to take the opening set.
The second set was similar, as world number 89 Hanescu broke in the third game and continued to serve well, eventually leading him to victory.
Hanescu won 75 per cent of points on his first serve throughout the match, and saved all five break points he offered his opponent, enough to book a place in the quarter-finals.
He will now face Federico Delbonis, who overcame Flilip Peliwo 6-2 6-7 (5-7) 6-4 in the day's other second round match.
The Argentinian won four games in a row to take the first set after scores were locked at 2-2, and he looked on his way to a comfortable victory after an early break in the second.
Canadian qualifier Peliwo broke back though and got the better of the tie-break to force a decider, although sixth seed Delbonis eventually proved too strong.
There was also a shock in the first round as Kazakh Aleksandr Nedovyesov beat seventh seed Robin Haase 7-6 (8-6) 6-4.
Dutchman Haase looked to be in complete control of the opening set when he held a 5-2 lead, but Nedovyesov roared back to force a tie-break, in which he came from 6-4 down to further display his battling qualities.
The second set went largely on serve, before a late break helped Nedovyesov claim an upset win.
He will be joined in the second round by eighth seed Guillermo Garcia-Lopez, with the Spaniard needing two tie-breaks to see off another Kazakh, Mikhail Kukushkin, 7-6 (7-5) 7-6 (8-6).
Garcia-Lopez's compatriot Albert Montanes also enjoyed a straight-sets victory, getting the better of Kenny de Schepper of the Netherlands 6-2 6-3.
Elsewhere, Czech Jiri Vesely beat Argentine Leonardo Mayer 6-4 6-3 and Spain's Pablo Carreno-Busta beat German Tobias Kamke, 6-1 6-2.