Jamie Buchan produced a wonderful first-half strike to put Dundee United on course to a vital three points against Motherwell.
United started confidently against Motherwell and after just 40 seconds a determined run by Jim Hamilton drew Benito Kemble into making a clumsy tackle.
But the home side could not make anything of the free-kick.
United were still looking nervous and David Partridge needlessly put his side under pressure with a loose pass-back to goalkeeper Paul Gallacher, which he was unable to keep in.
Motherwell forced three early corners, but United were managing to keep the inexperienced visitors at bay.
Again United showed promise in attack and Ange Oueifio obstructed David Hannah after a great ball from Miller, but again they were unable to make anything from the free-kick in a dangerous area.
United continued to grow in confidence and in the 13th minute Danny Griffin fired a right-foot free-kick from 20 yards over the crossbar, after Oueifio had brought down Hamilton.
A minute later and Motherwell youngster Keith Lasley was fortunate to escape a booking after cutting down Hannah but again United were wasteful from a dangerous position.
However Motherwell created the best chance of the match so far in the 16th minute when Said Chiba knocked the ball down-field to Kevin Twaddle and the skipper's first-time strike flew just wide of the upright.
Connolly was tested for the first time in the 22nd minute as Derek Lilley's looping effort from Miller's corner had the youngster frantically back-pedalling but he managed to catch the ball just underneath his own crossbar.
Chiba looked impressive with some good passes and looked most likely to create the opener for Motherwell.
But a clever through-ball from Easton in the 32nd minute almost created a good chance for Jamie Buchan, who had made a good run into the box, but the youngster's control let him down and the chance went begging.
The former Aberdeen player, however, made amends in the 37th minute when he fired his side in front.
Buchan picked up the ball on the half-way line before breaking forward and unleashing a swerving right-foot drive from long range which flew past Connolly and nestled in the top corner.
United grew in confidence from that goal and they should have doubled their lead four minutes later.
Hamilton's dummy from Easton's pass gave Lilley a great chance but he fired his right-foot effort straight at Connolly.
Connolly again kept his side in the game right on the stroke of half-time as he saved Partridge's fierce left-foot volley.