Denny Lucas finished alone in fourth at 4-under-par 67. He was followed by Sam Little, Bjorn Pettersson, Philippe Lima, Federico Bisazza, Joakim Rask, Alvaro Salto and Garry Houston at 3-under-par 68.
Rumford parred his first five holes on the Val Course at Aa Saint Omer Golf Club before picking up a birdie at the par-4 sixth. He then parred the seventh before going on a remarkable stretch to surge to the top of the leaderboard.
The 25-year-old ran off five consecutive birdies starting at the par-4 eight. Rumford added his seventh birdie of the day at the par-5 14th to secure sole possession of first after round one at the former Challenge Tour event.
Rumford, who won the ANZ Championship in his native Australia as an amateur in 1999, is in the process of regaining his European Tour card.
'I had a few changes last year,' said Rumford. 'This game is more that just teeing it up and going out and playing. Your game starts to go backwards as you are not mentally geared up to play and then you get a snowball effect of negativity that goes through your head.'
Monasterio played the back side first and opened with a birdie at the 10th. He then birdied the 14th but found trouble with a bogey at the 18th to make the turn at 1-under.
The Argentine played a brilliant inward half with four birdies to complete a round of 66.
Eliasson also tallied six birdies and one bogey for his share of second.
'I'm very pleased with that,' said Eliasson. 'This is a big opportunity as someone will get a year exemption. I like this course and finished fourth here last year so I feel I know it well. A couple of spots better than last year would be nice.'
Marc Pendaries, Simon Dyson, Michael Kirk, Sion E. Bebb, Peter Hanson, Paul Broadhurst and Sebastien Branger share 12th place at 2-under-par 69.
Nicolas Vanhootegem, who won this event last year, finished in a group at 3-over-par 74.