Allen beat Peter Ebdon in the opening match in Banbury, while Robertson came from behind to all-but end Shaun Murphy's semi-final hopes.
China Open winner Ebdon took the lead in the opening match, winning a scrappy first frame 72-34 but World Open champion Allen responded with a fine 84 break in the second to level.
Ebdon had the third frame at his mercy when leading 30-5 but left an easy red over the corner pocket and Allen capitalised, knocking in a 76 break to triumph 81-30 and seize the initiative in the match.
The Pistol's good form continued in frame four with a 50 break as he won 75-1 to establish a two frame lead.
But the Force refused to bow down and notched a 108 break in the fifth frame to keep himself in the hunt for a point.
But Allen was not to be denied and crafted a 69 break in the final frame to take the points and jump up to second in the Group B table behind Ebdon.
"I started slowly, feeling a bit nervous out there. But I think after that I settled down and played pretty well," Allen told Sky Sports 3HD afterwards.
In the second match of the evening, Masters champion Robertson also beat Murphy 4-2 to maintain his perfect start in Group A.
But the Australian, who beat Mark Selby 5-1 in his opening match, had to twice come from behind to earn the points.
Murphy, who had just one point from his opening two matches, notched a break of 25 at the death to take a scrappy first frame 60-44.
A tight second frame then hinged on the pink which Robertson potted to win 73-58 to level the match.
Murphy bounced back with an excellent 111 break in the third frame to re-establish his lead but again Robertson drew level with a 70 break in frame four.
The Australian then hit a superb 113 break in the penultimate frame to go ahead for the first time in the match.
Murphy established a 21-point lead in the final frame but suffered a bad kick on a simple red and the in-form Robertson took advantage, notching a 58 break to take the victory and open up a two point lead at the top of Group A.