The 28-year-old had a measured start to the match as he eased his way into action on Centre Court, but then took the first set with a break at the perfect time to clinch the first set 6-4.
Murray continued his momentum in the second set as he broke Kukushkin in his first service game, but the Kazakh showed resolve to break back, although the Scot edged ahead once more with another fine display against Kukushkin's serve.
The 27-year-old began to find rhythm on his forehand to take the game to Murray and powered back into the set to take a 6-5 lead. However, the number three seed discovered his best form to break back and take the set 7-3 on the tie-breaker.
Murray secured an early break of serve in the third set and held off the desperate attempts of the Kazakh to serve for the match, and despite conceding two break points, the Scot found his form and secured the victory.